**Basic Real Analysis on External Non-Archimedean Field $\mathbb{R}_c^#$.**
**Basic Complecs Analysis on External Field $C_c^# = \mathbb{R}_c^#[i]$.**

Jaykov Foukzon

jaykovfoukzon@gmail.com

Israel Institute of Technology, Haifa, Israel.

**Abstract:** In this paper we deal with set theory $\text{NC}_\infty^#$ based on gyper infinitary logic with Restricted Modus Ponens Rule [1]-[3].

The main goal of this paper is to present basic analysis on external non Archimedean field $\mathbb{R}_c^#$.

The non Archimedean external field $\mathbb{R}_c^#$ consist of Cauchy hyperreals.The non Archimedean external field $\mathbb{R}_c^# \neq *\mathbb{R}$ is obtained as generalized Cauchy completion of non-Archimedean field $\mathbb{Q}^#$ or $*\mathbb{Q}$. In order to obtain such completion we deal with external hyper infinite Cauchy sequences $\{x_n\}_{n \in \mathbb{N}^#}$, $\{x_n\}_{n \in *\mathbb{N}}$. We have emphasised that such external Cauchy sequences defined external hyperreal numbers in natural way. Basic Analysis on External Non-Archimedean Field $\mathbb{R}_c^#$ is considered.

Keywords: Infinitary logic;non Achromedian field.

2010 Mathematics Subject Classification: 03B10, 03B53, 03B60, 03E70, 03E65.

**Contents**

**Part I.Basic Analysis on External Non-Archimedean Field $\mathbb{R}_c^#$.**

1.Introduction

2. Set Theory $\text{NC}_\infty^#$ Based on Gyper Infinitary Logic with
Restricted Modus Ponens Rule.

3. Hypernaturals $\mathbb{N}^#$.

4. Hyper inductive definitions in general.

5. Fundamental examples of the hyper inductive definitions.

6. Nonstandard arithmetic $\mathbb{A}^#$.

7. Hyperrationals $\mathbb{Q}^#$.

8. External Cauchy hyperreals $\mathbb{R}_c^#$ via Cauchy completion of $\mathbb{Q}^#$.

9. The Extended Hyperreal Number System $\hat{\mathbb{R}}_c^#$.

10. External non-Archimedean field $*\mathbb{R}_c^#$.

10.1. External non-Archimedean field $*\mathbb{R}_c^#$ via Cauchy completion of $*\mathbb{R}$.

10.2. The Extended Hyperreal Number System $*\hat{\mathbb{R}}_c^#$.

10.3. External Cauchy hyperreals $*\mathbb{R}_c^#$ and $\mathbb{R}_c^#$ axiomatically.
## 11. Basic analysis on External non-Archimedean field $\mathbb{R}_c^#$.

11.1. The #-limit of a function $f: \mathbb{R}_c^# \to \mathbb{R}_c^#$.

11.2. Monotonic External Functions $f: \mathbb{R}_c^# \to \mathbb{R}_c^#$.

11.3. #-Limits Inferior and Superior

11.4. The #-continuity of a function $f: \mathbb{R}_c^# \to \mathbb{R}_c^#$.

11.5. Removable #-discontinuities.

11.6. Composite Functions $f: \mathbb{R}_c^# \to \mathbb{R}_c^#$.

11.7. Bounded and Hyperbounded Functions $f: \mathbb{R}_c^# \to \mathbb{R}_c^#$.

11.8. Intermediate Value Theorem.

11.9. Uniform #-Continuity

11.10. Monotonic External Functions $f: \mathbb{R}_c^# \to \mathbb{R}_c^#$.

11.11. The #-derivative of a $\mathbb{R}_c^#$-valued function $f: D \to \mathbb{R}_c^#$.

11.12. The Riemann integral of a $\mathbb{R}_c^#$-valued external function $f(x)$.

11.13. Existence of the integral of a $\mathbb{R}_c^#$-valued external function $f(x)$.

11.14. Improper integrals of a $\mathbb{R}_c^#$-valued external function $f(x)$.

11.15. Improper integrals of nonnegative functions $f: D \to \mathbb{R}_c^#$. Absolute Integrability.

11.16. Change of Variable in Improper Integrals of a $\mathbb{R}_c^#$-valued external function.

11.17. Generalized integrability criterion of a $\mathbb{R}_c^#$-valued external function $f(x)$ due to Lebesgue.

## 12. Hyper infinite external sequences and series of constants.

12.1. Hyper infinite external sequences of constants.

12.2. Hyper infinite external series of constants.

12.3. Hyper infinite external series of nonnegative terms.

12.4. Absolute and Conditional #-Convergence.

12.5. Grouping Terms in a Hyper Infinite Series.

12.6. Rearrangement of hyper infinite series.

12.7. Multiplication of hyper infinite Series

12.8. Double Hyper Infinite Sequences.

12.9. Double Hyper Infinite Series.

12.10. Interchanging the Order of Summation of Hyper Infinite Sum.

## 13. Hyper infinite sequences and series of $\mathbb{R}_c^#$-valued functions

13.1. Uniform #-Convergence.

13.2. Hyper Infinite Series of Functions.

13.3. #-Continuity, #-Differentiability, and Integrability of hyper infinite Series.

## 14. Hyper Infinite Power Series.

14.1. The #-convergence properties of hyper infinite power series.

14.2. The $\mathbb{R}_c^#$-valued #-exponential $Ext-\exp(x)$

14.3. The $\mathbb{R}_c^#$-valued Trigonometric Functions $Ext-\sin(x)$, $Ext-\cos(y)$.

14.4. $\mathbb{R}_c^#$-valued functions of several variables.

14.5. Line Segments in $\mathbb{R}_c^{\#n}, n \in \mathbb{N}^#$.

14.6. Neighborhoods and #-open sets in $\mathbb{R}_c^{\#n}, n \in \mathbb{N}^#$. Connected Sets and Regions in $\mathbb{R}_c^{\#n}$.

14.7. The #-limits and #-continuity $\mathbb{R}_c^#$-valued functions of $n \in \mathbb{N}^#$ variables.

14.8. Partial #-Derivatives and the #-Differential.

## 15. #-analytic functions $f: \mathbb{C}_c^# \to \mathbb{C}_c^#$.

15.1. $\mathbb{C}_c^#$-valued #-analytic functions $f: \mathbb{C}_c^# \to \mathbb{C}_c^#$.
15.2. The $C_c^#$-valued $\#$-exponential $Ext-\exp(z)$.

15.3. The $C_c^#$-valued Trigonometric Functions $Ext-\sin(z)$, $Ext-\cos(z)$.

15.4. The Periodicity of the $\#$-exponential $Ext-\exp(iz)$.

15.5. The $C_c^#$-valued Logarithm

**16. Complex Integration of the $C_c^#$-valued function $f(t)$.**

16.1. Definition and basic properties of the complex integral.

16.2. Line Integrals as Functions of Arcs.

16.3. Generalized Cauchy's Theorem for a Rectangle.

16.4. Generalized Cauchy's Theorem in a Disk.

16.5. Generalized Cauchy's integral formula.

16.6. Generalized Liouville's theorem.

16.7. Generalized fundamental theorem of algebra.

**17. Local Properties of $\#$-Analytical Functions**

17.1. Removable Singularity. Generalized Taylor's Theorem.

17.2. Zeros and Poles of $\#$-Analytical Functions.

17.3. The Generalized Maximum Principle.

**Part II. External Lebesgue Integral of the $R_c^#$-valued function $f(t)$.**

**1. External Lebesgue Measure on $R_c^#**

1.1. External Lebesgue outer measure.

1.2. External Lebesgue measure.

**2. External Lebesgue Integral**

2.1. External Lebesgue Integral of a $R_c^#$-valued external function $f(x)$.

2.2. General Definition and Basic Properties of the external Lebesgue Integral.

**Appendix A.** Bivalent Hyper Infinitary first-order logic $^2L_\infty^#$ with restricted rules of conclusion. Generalized Deduction Theorem.

**Appendix B.** The Generalized Recursion Theorem.

**Appendix C.** General associative and commutative laws.

**1. Introduction**

In this paper we deal with set theory $NC_\infty^#$ based on hyper infinitary logic with
Restricted Modus Ponens Rule [1]-[3].

The main goal of this paper is to present basic analysis on non Archimedean field $R_c^#$.

The non Archimedean field $R_c^#$ consist of Cauchy hyperreals. The non Archimedean
field $R_c^#$ is obtained as generalized Cauchy completion of non Archimedean field $Q^#$
or $*Q$. In order to obtain such completion we deal with external hyper infinite Cauchy
sequences {$x_n$}$_{n \in N^#}$, {$x_n$}$_{n \in |*N|}$.

Note that analysis on a non-Archimedean field $R_c^#$ is essentially different in comparison
with analysis on non Archimedean field $*\mathbb{R}$ [4]-[5] known in literature as nonstandard
analysis, see for example [4]-[5].

Remind that Robinson nonstandard analysis (RNA) many developed using
set-theoretical objects called superstructures [5]. A superstructure $V(S)$ over a set $S$ is
defined in the following way:
$$V_0(S) = S, V_{n+1}(S) = V_n(S) \cup (P(V_n(S)), V(S) = \bigcup_{n \in \mathbb{N}} V_n(S)). \quad (1.1)$$

Superstructures of the empty set consist of sets of infinite rank in the cumulative hierarchy and therefore do not satisfy the in...nity axiom. Making $S = \mathbb{R}$ will suffice for virtually any construction necessary in analysis.

Bounded formulas are formulas where all quantifiers occur in the form

$$\forall x(x \in y \Rightarrow \cdots), \exists x(x \in y \Rightarrow \cdots). \qquad (1.2)$$

A nonstandard embedding is a mapping

$$*: V(X) \to V(Y) \qquad (1.3)$$

from a superstructure $V(X)$ called the standard universum, into another superstructure $V(Y)$, called nonstandard universum, satisfying the following postulates:

**1.** $Y = *X$

**2. Transfer Principle.** For every bounded formula $\Phi(x_1, \dots, x_n)$ and elements $a_1, \dots, a_n \in V(X)$, the property $\Phi$ is true for $a_1, \dots, a_n$ in the standard universum if and only if it is true for $*a_1, \dots, *a_n$ in the nonstandard universum:

$$\langle V(X), \in \rangle \models \Phi(a_1, \dots, a_n) \Leftrightarrow \langle V(Y), \in \rangle \models \Phi(*a_1, \dots, *a_n). \qquad (1.4)$$

**3. Non-triviality.** For every infinite set $A$ in the standard universum, the set $\{*a|a \in A\}$ is a proper subset of $*A$.

**Definition 1.1.** [5]. A set $x$ is *internal* if and only if $x$ is an element of $*A$ for some element $A$ of $V(\mathbb{R})$. Let $X$ be a set with $A = \{A_i\}_{i \in I}$ a family of subsets of $X$. Then the collection $A$ has the infinite intersection property, if any infinite subcollection $J \subset I$ has non-empty intersection. Nonstandard universum is $\kappa$-saturated if whenever $\{A_i\}_{i \in I}$ is a collection of internal sets with the infinite intersection property and the cardinality of $I$ is less than or equal to $\kappa$, $\bigcap_{i \in I} A_i \neq \emptyset$.

**Definition 1.2.** [2]-[3]. A set $S \subset *\mathbb{N}$ is a *hyper inductive* if the following statement holds

$$\bigwedge_{\alpha \in *\mathbb{N}} (\alpha \in S \Rightarrow \alpha^+ \in S), \qquad (1.5)$$

where $\alpha^+ \triangleq \alpha + 1$. Obviously a set $*\mathbb{N}$ is a hyper inductive. As we see later there is just one hyper inductive subset of $*\mathbb{N}$, namely $*\mathbb{N}$ itself.

In this paper we apply the following hyper inductive definitions of a sets [2]-[3]

$$\exists S \forall \beta \left[ \beta \in S \Leftrightarrow \bigwedge_{0 \le \alpha < \beta} (\alpha \in S \Rightarrow \alpha^+ \in S) \right], \qquad (1.6)$$

We extend up Robinson nonstandard analysis (RNA) by adding the following postulate:

**4.** Any hyper inductive set $S$ is internal.

**Remark 1.1.** The statement **4** is not provable in ZFC but provable in set theory $\mathbb{NC}^{\#}_{\infty}$, see [2]-[3]. Thus postulates 1-4 gives an nonconservative extension of RNA and we denote such extension by NERNA.

**Remark 1.2.** Note that NERNA of course based on the same hyper infinitary logic with Restricted Modus Ponens Rule as set theory $\mathbb{NC}^{\#}_{\infty}$ [1]-[3]. Remind that in RNA the following induction principle holds.
**Theorem 1.1.**[6]. Assume that $S \subset *N$ is internal set, then

$$ (1 \in S) \wedge \forall x[x \in S \Rightarrow x + 1] \Rightarrow S = *N. \qquad (1.7) $$

In NERNA Theorem 1.1 also holds.

**Remark 1.3.** It follows from postulate 4 and Theorem 1.1 that any hyper inductive set $S$ is equivalent to $*N : S = *N$.

**Remark 1.4.** Note that the following statement is provable in $NC_{\infty}^{\#}$ [2-3]:

**4' Axiom of hyper infinite induction**

$$ \forall S(S \subset *N) \left\{ \forall \beta(\beta \in *N) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in S \Rightarrow \alpha^{+} \in S) \right] \Rightarrow S = *N \right\}. \qquad (1.8) $$

Thus postulate **4** of the theory NERNA is provable in $NC_{\infty}^{\#}$.

**Remark 1.5.** Note that $*N$ is an model related to hypernaturals $N^{\#}$ which we introduce axiomatically in [3], see subsection 2.1.

The paper is structured as follows.

In Sec. 2 set theory $NC_{\infty}^{\#}$ is formulated as a system of axioms based on bivalent hyper infinitary logic $2L_{\infty}^{\#}$ with restricted modus ponens rule [1]-[3],[9]. In Subsec.2.1. Axiom of nonregularity and axiom of hyperinfinity is formulated.

In Sec.3 nonstandard arithmetic $A^{\#}$ related to hypernaturals $N^{\#}$ [2-3] is formulated axiomatically.

In Sec.4 hyper inductive definitions in general case is considered.

In Sec.5. Fundamental examples of the hyper inductive definitions is considered.

In Sec.6. Nonstandard arithmetic $A^{\#}$ is formulated by using finitary logic.

In Sec.7 defining hyperintegers $Z^{\#}$ and hyperrational $Q^{\#}$ numbers are given [2].

In Sec.8 Cauchy hyperreals $\mathbb{R}_{\epsilon}^{\#}$ via generalized Cauchy completion is formulated.

In Sec.9 Extended Hyperreal Number System $\mathbb{R}_{\epsilon}^{\#}$ is considered.

In Sec.11 Basic analysis on external non Archimedean field $\mathbb{R}_{\epsilon}^{\#}$ is considered.

## 2. Set Theory $NC_{\infty}^{\#}$ Based on Bivalent Hyper Infinitary Logic with Restricted Modus Ponens Rule.

Set theory $NC_{\infty}^{\#}$ is formulated as a system of axioms based on bivalent hyper infinitary logic $2L_{\infty}^{\#}$ with restricted modus ponens rule [1]-[3], see Appendix A. The language of set theory $NC_{\infty}^{\#}$ is a first-order hyper infinitary language $L_{\infty}^{\#}$ with equality $=$, which includes a binary symbol $\in$. We write $x \neq y$ for $\neg (x = y)$ and $x \notin y$ for $\neg (x \in y)$. Individual variables $x, y, z, \dots$ of $L_{\infty}^{\#}$ will be understood as ranging over classical sets. The unique existential quantifier $\exists!$ is introduced by writing, for any formula $\varphi(x), \exists!x\varphi(x)$ as an abbreviation of the formula $\exists x[\varphi(x) \ \& \ \forall y(\varphi(y) \Rightarrow x = y)]$. $L_{\infty}^{\#}$ will also allow the formation of terms of the form $\{x|\varphi(x)\}$, for any formula $\varphi$ containing the free variable $x$. Such terms are called non-classical sets; we shall use upper case
letters A, B, ... for such sets. For each non-classical set $A = \{x|\varphi(x)\}$ the formulas $\forall x[x \in A \Leftrightarrow \varphi(x)]$ and $\forall x[x \in A \Leftrightarrow \varphi(x,A)]$ is called the defining axioms for the non-classical set A.

**Remark 2.1.** Remind that in logic $^{2}L_{\infty}^{\#}$ with restricted modus ponens rule the statement $\alpha \wedge (\alpha \Rightarrow \beta)$ does not always guarantee that

$$ \alpha, \alpha \Rightarrow \beta \vdash_{RMP} \beta \qquad (2.1) $$

since for some $\alpha$ and $\beta$ possible

$$ \alpha, \alpha \Rightarrow \beta \dashv_{RMP} \beta \qquad (2.2) $$

even if the statement $\alpha \wedge (\alpha \Rightarrow \beta)$ holds [1].

**Abbreviation 2.2.** We often write for the sake of brevity instead (2.1) by

$$ \alpha \Rightarrow_s \beta \qquad (2.3) $$

and we often write instead (2.2) by

$$ \alpha \Rightarrow_w \beta. \qquad (2.4) $$

**Remark 2.2.** Let A be an nonclassical set. Note that in set theory $NC_{\infty}^{\#}$ the following true formula

$$ \exists A \forall x[x \in A \Leftrightarrow \varphi(x,A)] \qquad (2.5) $$

does not always guarantee that

$$ x \in A, x \in A \Rightarrow \varphi(x,A) \vdash_{RMP} \varphi(x,A) \qquad (2.6) $$

even if $x \in A$ holds and (or)

$$ \varphi(x,A), \varphi(x,A) \Rightarrow x \in A \vdash_{RMP} x \in A; \qquad (2.7) $$

even $\varphi(x,A)$ holds, since for nonclassical set A for some y possible

$$ y \in A, y \in A \Rightarrow \varphi(y,A) \dashv_{RMP} \varphi(y,A) \qquad (2.8) $$

and (or)

$$ \varphi(y,A), \varphi(y,A) \Rightarrow y \in A \dashv_{RMP} y \in A. \qquad (2.9) $$

**Remark 2.3.** Note that in this paper the formulas

$$ \exists a \forall x[x \in a \Leftrightarrow \varphi(x) \wedge x \in u] \qquad (2.10) $$

and more general formulas

$$ \exists a \forall x[x \in a \Leftrightarrow \varphi(x,a) \wedge x \in u] \qquad (2.11) $$

is considered as the defining axioms for the classical set a.

**Remark 2.4.** Let a be a classical set. Note that in $NC_{\infty}^{\#}$: (i) the following true formula

$$ \exists a \forall x[x \in a \Leftrightarrow \varphi(x,a) \wedge x \in u] \qquad (2.12) $$

always guarantee that

$$ x \in a, x \in a \Rightarrow \varphi(x,a) \vdash_{RMP} \varphi(x) \qquad (2.13) $$

if $x \in a$ holds and

$$ \varphi(x), \varphi(x) \Rightarrow x \in a \vdash_{RMP} x \in a; \qquad (2.14) $$

if $\varphi(x)$ holds;

In order to emphasize this fact mentioned above in Remark 2.1-2.3,
we rewrite the defining axioms in general case for the nonclassical sets in the
following
form

$$
\exists A \forall x \{[x \in A \Leftrightarrow_s \varphi(x, A)] \lor [x \in A \Leftrightarrow_w \varphi(x, A)]\} \quad (2.15)
$$

and similarly we rewrite the defining axioms in general case for the classical sets in
the
following form

$$
\forall x[x \in a \Leftrightarrow_s \varphi(x, a) \wedge (x \in u)]. \quad (2.16)
$$

**Abbreviation 2.2.** We write instead (2.15):

$$
\forall x \{[x \in A \Leftrightarrow_{s,w} \varphi(x,A)]\} \qquad (2.17)
$$

**Definition 2.1.** (1) Let A be a nonclassical set defined by formula (2.17).
Assum that: (i) for some y statement φ(y) and statement φ(y) ⇒ y ∈ A holds and
(ii) φ(y), φ(y) ⇒ y ∈ A ⊢RMP y ∈ A, y ∈ A, y ∈ A ⇒ φ(y) ⊢RMP φ(y).
Then we say that y is a weak member of non-classical set A and abbreviate y ∈<sub>w</sub> A.

**Abbreviation 2.3.** Let $A$ be a nonclassical set defined by formula (6.1) or by formula (6.2). We abbreviate $x \in_{s,w} A$ if the following statement $x \in_s A \lor x \in_w A$ holds, i.e.

$$
x \in_{s,w} A \leftrightarrow_{def} (x \in_s A \lor x \in_w A). \quad (2.18)
$$

**Definition 2.2.** (1) Two nonclassical sets A, B are defined to be equal and we write
A = B if $\forall x[x \in_{s,w} A \Leftrightarrow_s x \in_{s,w} B]$. (2) A is a subset of B, and we often write $A \subset_{s,v} B$, if
$\forall x[x \in_{s,w} A \Rightarrow_s x \in_{s,w} B]$. (3) We also write **CL. Set**(A) for the formula
$\exists u \forall x[x \in A \Leftrightarrow x \in u]$. (4) We also write **NCI. Set**(A) for the formulas
$\forall x[x \in_{s,v} A \Leftrightarrow_{s,v} \varphi(x)]$ and $\forall x[x \in_{s,v} A \Leftrightarrow_{s,v} \varphi(x, A)]$.

**Remark 2.5.** CL. Set(A)) asserts that the set A is a classical set. For any classical set u,
it follows from the defining axiom for the classical set {x|x ∈ u ∧ φ(x)} that
CL. Set( {x|x ∈ u ∧ φ(x)}) .

We shall identify {$x|x \in_s u$} with $u$, so that sets may be considered as (special sorts of)
nonclassical sets and we may introduce assertions such as $u \subset_s A, u \subseteq_s A$, etc.

**Abbreviation 2.4.** Let $\varphi(t)$ be a formula of $NC^#_{\infty}$.

(i) $\forall x \varphi(x)$ and $\forall^{\text{CL}} x \varphi(x)$ abbreviates $\forall x (\text{CL. Set}(x) \Rightarrow \varphi(x))$

(ii) $\exists x \varphi(x)$ and $\exists^{\text{CL}} x \varphi(x)$ abbreviates $\forall x (\text{CL. Set}(x) \Rightarrow \varphi(x))$

(iii) $\forall X \varphi(X)$ and $\forall^{\text{NCL}} X \varphi(X)$ abbreviates $\forall X (\text{NCL. Set}(X) \Rightarrow \varphi(X))$

(iv) $\exists X \varphi(X)$ and $\exists^{\text{NCL}} X \varphi(X)$ abbreviates $\exists X (\text{NCL. Set}(X) \Rightarrow \varphi(X))$

**Remark 2.6.** If A is a nonclassical set, we write $\exists x \in A \varphi(x, A)$ for $\exists x [x \in A \land \varphi(x, A)]$ and $\forall x \in A \varphi(x, A)$ for $\forall x [x \in A \Rightarrow \varphi(x, A)]$.

We define now the following sets:

1. $\{u_1, u_2, \dots, u_n\} = \{x|x = u_1 \lor x = u_2 \lor \dots \lor x = u_n\}$. 2. $\{A_1, A_2, \dots, A_n\} = \{x|x = A_1 \lor x = A_2 \lor \dots \lor x = A_n\}$. 3. $\cup A = \{x|\exists y[y \in A \land x \in y]\}$.

4. $\cap A = \{x | \forall y [y \in A \Rightarrow x \in y]\}$. 5. $A \cup B = \{x | x \in A \lor x \in B\}$.

5. $A \cap B = \{x|x \in A \land x \in B\}$. 6. $A - B = \{x|x \in A \land x \notin B\}$. 7. $u^+ = u \cup \{u\}$.

8. $\mathbf{P}(A) = \{x|x \subseteq A\}$. 9. $\{x \in A | \varphi(x, A)\} = \{x|x \in A \land \varphi(x, A)\}$. 10. $\mathbf{V} = \{x: x = x\}$.

11. $\emptyset = \{x|x \neq x\}$.

The system $NC^#_{\infty}$ of set theory is based on the following axioms:

**Extensionality**1: $\forall u \forall v [\forall x (x \in u \Leftrightarrow x \in v) \Rightarrow u = v]$
**Extensionality**2: $\forall A \forall B [\forall x (x \in A \Leftrightarrow_{s,w} x \in B) \Rightarrow A = B]$

**Universal Set:** NCL. Set(V)

**Empty Set:** CL. Set(Ø)

**Pairing**1: $\forall u \forall v \text{ CL. Set}(\{u, v\})$

**Pairing**2: $\forall A \forall B \text{ NCL. Set}(\{A, B\})$

**Union**1: $\forall u \text{ CL. Set}(\cup u)$

**Union**2: $\forall A \text{ NCL. Set}(\cup A)$

**Powerset**1: $\forall u \text{ CL. Set}(P(u))$

**Powerset**2: $\forall A \text{ NCL. Set}(P(A))$

**Infinity:** $\exists a[\emptyset \in a \land \forall x \in a(x^+ \in a)]$

**Separation**1 $\forall u_1 \forall u_2, \dots \forall u_n \forall a \exists \text{CL. Set}(\{x \in s a | \varphi(x, u_1, u_2, \dots, u_n)\})$

**Separation**2 $\forall u_1 \forall u_2, \dots \forall u_n \text{ NCL. Set}(\{x \in s,w A | \varphi(x, A; u_1, u_2, \dots, u_n)\})$

**Comprehension**1 $\forall u_1 \forall u_2, \dots \forall u_n \exists A \forall x [x \in s,w A \Leftrightarrow_{s,w} \varphi(x; u_1, u_2, \dots, u_n)]$

**Comprehension** 2 $\forall u_1 \forall u_2, \dots \forall u_n \exists A \forall x [x \in s,w A \Leftrightarrow_{s,w} \varphi(x, A; u_1, u_2, \dots, u_n)]$

**Comprehension** 3 $\forall u_1 \forall u_2, \dots \forall u_n \exists a \forall x [x \in s a \Leftrightarrow_s (a \subset u_1) \land \varphi(x, a; u_1, u_2, \dots, u_n)]$

In particular:

**Comprehension** 3' $\forall u \exists a \forall x [x \in s a \Leftrightarrow_s (a \subset u) \land \varphi(x, a; u)]$

**Hyperinfinity:** see subsection 2.1.

**Remark 2.7.** Note that the axiom of hyper infinity follows from the schemata Comprehension 3.

**Definition 2.3.** The ordered pair of two sets $u, v$ is defined as usual by

$$ \langle u, v \rangle = \{\{u\}, \{u, v\}\}. \qquad (2.19) $$

**Definition 2.4.** We define the Cartesian product of two nonclassical sets A and B as usual by

$$ A \times_{s,w} B = \{\langle x, y \rangle | x \in_{s,w} A \land y \in_{s,w} B \} \qquad (2.20) $$

**Definition 2.5.** A binary relation between two nonclassical sets A, B is a subset $R \subseteq_{s,w} A \times_{s,w} B$. We also write $aR_{s,w}b$ for $\langle a, b \rangle \in_{s,w} R$. The doman $\text{dom}(R)$ and the range $\text{ran}(R)$ of R are defined by

$$ \text{dom}(R) = \{x | \exists y (xR_{s,w}y)\}, \text{ran}(R) = \{y : \exists x (xR_{s,w}y)\}. \qquad (2.21) $$

**Definition 2.6.** A relation $F_{s,w}$ is a function, or map, written $\text{Fun}(F_{s,w})$, if for each $a \in_{s,w} \text{dom}(F)$ there is a unique $b$ for which $aF_{s,w}b$. This unique $b$ is written $F(a)$ or $Fa$. We write $F_{s,w}: A \to B$ for the assertion that $F_{s,w}$ is a function with $\text{dom}(F_{s,w}) = A$ and $\text{ran}(F_{s,w}) = B$. In this case we write $a \mapsto F_{s,w}(a)$ for $F_{s,w}a$.

**Definition 2.7.** The identity map $1_A$ on A is the map $A \to A$ given by $a \mapsto a$. If $X \subseteq_{s,w} A$, the map $x \mapsto x: X \to A$ is called the insertion map of X into A.

**Definition 2.8.** If $F_{s,w}: A \to B$ and $X \subseteq_{s,w} A$, the restriction $F_{s,w}|_X$ of $F_{s,w}$ to X is the map $X \to A$ given by $x \mapsto F_{s,w}(x)$. If $Y \subseteq_{s,w} B$, the inverse image of Y under $F_{s,w}$ is the set

$$ F_{s,w}^{-1}[Y] = \{x \in_{s,w} A : F_{s,w}(x) \in_{s,w} Y\}. \qquad (2.22) $$

Given two functions $F_{s,w}: A \to B, G_{s,w}: B \to C$, we define the composite function

$G_{s,w} \circ F_{s,w}: A \to C$ to be the function $a \mapsto G_{s,w}(F_{s,w}(a))$. If $F_{s,w}: A \to A$, we write $F_{s,w}^2$ for $F_{s,w} \circ F_{s,w}, F_{s,w}^3$ for $F_{s,w} \circ F_{s,w} \circ F_{s,w}$ etc.

**Definition 2.9.** A function $F_{s,w}: A \to B$ is said to be monic if for all
$x, y \in_{s,w} A, F_{s,w}(x) = F_{s,w}(y)$ implies $x = y$, epi if for any $b \in_{s,w} B$ there is $a \in_{s,w} A$ for which $b = F_{s,w}(a)$, and bijective, or a bijection, if it is both monic and epi. It is easily shown that

$F_{s,w}$ is bijective if and only if $F_{s,w}$ has an inverse, that is, a map $G_{s,w} : B \to A$ such that $F_{s,w} \circ G_{s,w} = \mathbf{1}_B$ and $G_{s,w} \circ F_{s,w} = \mathbf{1}_A$.

**Definition 2.10.** Two sets $X$ and $Y$ are said to be equipollent, and we write $X \approx_{s,w} Y$, if there is a bijection between them.

**Definition 2.11.** Suppose we are given two sets $I, A$ and an epi map $F_{s,w} : I \to A$. Then $A = \{F_{s,w}(i) | i \in I\}$ and so, if, for each $i \in_{s,w} I$, we write $a_i$ for $F_{s,w}(i)$, then $A$ can be presented in the form of an indexed set $\{a_i : i \in_{s,w} I\}$. If $A$ is presented as an indexed set of sets $\{X_i | i \in_{s,w} I\}$, then we write $\bigcup_{i \in I} X_i$ and $\bigcap_{i \in I} X_i$ for $\cup A$ and $\cap A$, respectively.

**Definition 2.12.** The projection maps $\pi_1 : A \times_{s,w} B \to A$ and $\pi_2 : A \times_{s,w} B \to B$ are defined to be the maps $\langle a, b \rangle \mapsto a$ and $\langle a, b \rangle \mapsto b$ respectively.

**Definition 2.13.** For sets $A, B$, the exponential $B^A$ is defined to be the set of all functions from $A$ to $B$.

## 2.1. Axiom of nonregularity and axiom of hyperinfinity

### Axiom of nonregularity

Remind that a non-empty set $u$ is called regular iff $\forall x[x \neq \emptyset \to (\exists y \in x)(x \cap y = \emptyset)]$.

Let's investigate what it says: suppose there were a non-empty $x$ such that $(\forall y \in x)(x \cap y \neq \emptyset)$. For any $z_1 \in x$ we would be able to get $z_2 \in z_1 \cap x$. Since $z_2 \in x$ we would be able to get $z_3 \in z_2 \cap x$. The process continues forever:
... $\in z_{n+1} \in z_n... \in z_4 \in z_3 \in z_2 \in z_1 \in x$. Thus if we don't wish to rule out such an infinite regress we forced accept the following statement:

$$ \exists x[x \neq \emptyset \to (\forall y \in x)(x \cap y \neq \emptyset)]. \qquad (2.23) $$

### Axiom of hyperinfinity.

**Definition 2.14.** (i) A non-empty transitive non regular set $u$ is a well formed non regular

set iff:

(i) there is unique countable sequence $\{u_n\}_{n=1}^{\infty}$ such that

$$ \dots \in u_{n+1} \in u_n \dots \in u_4 \in u_3 \in u_2 \in u_1 \in u, \qquad (2.24) $$

(ii) for any $n \in \mathbb{N}$ and any $u_{n+1} \in u_n$ :

$$ u_n = u_{n+1}^+, \qquad (2.25) $$

where $a^+ = a \cup \{a\}$.

(ii) we define a function $a^{+[k]}$ inductively by $a^{+[k]} = (a^{+[k]})^+$

**Definition 2.15.** Let $u$ and $w$ are well formed non regular sets. We write $w < u$ iff for any $n \in \mathbb{N}$

$$ w \in u_n. \qquad (2.26) $$

**Definition 2.16.** We say that a well formed non regular set $u$ is infinite (or hyperfinite) hypernatural nuber iff:
(I) For any member $w \in u$ one and only one of the following conditions are satisfied:

(i) $w \in \mathbb{N}$ or

(ii) $w = u_n$ for some $n \in \mathbb{N}$ or

(iii) $w < u$.

(II) Let $\prec u$ be a set $\prec u = \{z|z < u\}$, then by relation $(\cdot < \cdot)$ a set $\prec u$ is densely ordered with no first element.

(III) $\mathbb{N} \subset u$.

**Definition 2.17.** Assume $u \in \mathbb{N}^\#$, then $u$ is infinite (hypernatural) number if $u \in \mathbb{N}^\#\mathbb{N}$.

**Axiom of hyperinfinity**

There exists a set $\mathbb{N}^\#$ such that:

(i) $\mathbb{N} \subset \mathbb{N}^\#$

(ii) if $u \in \mathbb{N}^\#\mathbb{N}$ then there exists infinite (hypernatural) number $v$ such that $v < u$

(iii) if $u \in \mathbb{N}^\#\mathbb{N}$ then there exists infinite (hypernatural) number $w$ such that $u < w$

(v) set $\mathbb{N}^\#\mathbb{N}$ is patially ordered by relation $(\cdot < \cdot)$ with no first and no last element.

### 3. Hypernaturals $\mathbb{N}^\#$.

In this section nonstandard arithmetic $\mathbb{A}^\#$ related to hypernaturals $\mathbb{N}^\#$ is considered axiomatically.

**Axioms of the nonstandard arithmetic $\mathbb{A}^\#$ are:**

**Axiom of hyperinfinity**

There exists unique set $\mathbb{N}^\#$ such that:

(i) $\mathbb{N} \subset \mathbb{N}^\#$

(ii) if $u$ is infinite (hypernatural) number then there exists infinite (hypernatural) number $v$ such that $v < u$

(iii) if $u$ is infinite hypernatural number then there exists infinite (hypernatural) number $w$ such that $u < w$

(iv) set $\mathbb{N}^\#\mathbb{N}$ is patially ordered by relation $(\cdot < \cdot)$ with no first and no last element.

**Axioms of infinite $\omega$-induction**

(i)

$$ \forall S(S \subset \mathbb{N}) \left\{ \left[ \bigwedge_{n \in \omega} (n \in S \Rightarrow_s n^+ \in S) \right] \Rightarrow_s S = \mathbb{N} \right\}. \quad (3.1) $$

(ii) Let $F(x)$ be a wff of the set theory $\mathbb{N}\mathbb{C}_{\infty}^\#$, then

$$ \left[ \bigwedge_{n \in \omega} (F(n) \Rightarrow_s F(n^+)) \right] \Rightarrow_s \forall n(n \in \omega) F(n). \quad (3.2) $$

**Definition 3.1.** (i) Let $\beta$ be a hypernatural such that $\beta \in \mathbb{N}^\#\mathbb{N}$. Let $[0, \beta] \subset \mathbb{N}^\#$ be a set such that $\forall x[x \in [0, \beta] \Leftrightarrow 0 \le x \le \beta]$ and let $[0, \beta)$ be a set $[0, \beta) = [0, \beta]\backslash\{\beta\}$.

(ii) Let $\beta \in \mathbb{N}^\#\mathbb{N}$ and let $\beta_\infty \subset \mathbb{N}^\#$ be a set such that

$$ \forall x\{x \in \beta_{\infty} \Leftrightarrow \exists k(k \ge 0)[0 \le x \le \beta^{+[k]}]\}. \quad (3.3) $$

**Definition 3.2.** Let $F(x)$ be a wff of $\mathbb{N}\mathbb{C}_{\infty}^\#$ with unique free variable $x$. We will say that a wff $F(x)$ is restricted on a classical set $S$ such that $S \not\subseteq_s \mathbb{N}^\#$ iff the following condition is satisfied
$$ \forall \alpha[\alpha \in \mathbb{N}^{\#} \S \Rightarrow_s \neg F(\alpha)]. \qquad (3.4) $$

**Definition 3.3.** Let $F(x)$ be a wff of $\mathbf{NC}_{\infty}^{\#}$ with unique free variable $x$. We will say that a wff $F(x)$ is *strictly restricted* on a set $S$ such that $S \subseteq_s \mathbb{N}^{\#}$ iff there is no proper subset $S' \subset S$ such that a wff $F(x)$ is restricted on a set $S'$.

**Example 3.1.** (i) Let $\mathbf{fin}(\alpha)$, $\alpha \in \mathbb{N}^{\#}$ be a wff formula such that $\mathbf{fin}(\alpha) \Leftrightarrow_s \alpha \in \mathbb{N}$. Obviously wff $\mathbf{fin}(\alpha)$ is strictly restricted on a set $\mathbb{N}$ since $\forall \alpha[\alpha \in \mathbb{N}^{\#} \mathbb{N} \Rightarrow_s \neg \mathbf{fin}(\alpha)]$. Let $\mathbf{hfin}(\alpha)$, $\alpha \in \mathbb{N}^{\#}$ be a wff formula such that $\mathbf{hfin}(\alpha) \Leftrightarrow_s \alpha \in \mathbb{N}^{\#} \mathbb{N}$ since $\forall \alpha[\alpha \in \mathbb{N} \Rightarrow_s \neg \mathbf{hfin}(\alpha)]$.

**Definition 3.4.** Let $F(x)$ be a wff of $\mathbf{NC}_{\infty}^{\#}$ with unique free variable $x$. We will say that a wff $F(x)$ is unrestricted if wff $F(x)$ is not restricted on any set $S$ such that $S \subseteq \mathbb{N}^{\#}$.

**Axiom of hyperfinite induction 1**

$$ \forall S (S \subseteq_s [0, \beta]) \forall \beta (\beta \in_s \mathbb{N}^#) \searrow \\ \left\{ \forall \alpha (\alpha \in_s [0, \beta]) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in_s S \Rightarrow \alpha^+ \in_s S) \right] \Rightarrow_s S = [0, \beta] \right\}. \qquad (3.5) $$

**Axiom of hyperfinite induction 1'**

$$ \forall S (S \subseteq_s [0, \beta_{\infty}]) \forall \beta (\beta \in_s \mathbb{N}^#) \searrow \\ \left\{ \forall \alpha (\alpha \in [0, \beta_{\infty}]) \left[ \bigwedge_{0 \le \alpha < \beta_{\infty}} (\alpha \in S \Rightarrow \alpha^+ \in S) \right] \Rightarrow S = [0, \beta_{\infty}] \right\}. \qquad (3.5) $$

**Axiom of hyper infinite induction 1**

$$ \forall S (S \subseteq_s \mathbb{N}^#) \left\{ \forall \beta (\beta \in_s \mathbb{N}^#) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in_s S \Rightarrow \alpha^+ \in_s S) \right] \Rightarrow_s S =_s \mathbb{N}^# \right\}. \qquad (3.6) $$

**Definition 3.5.** A set $S \subseteq_s \mathbb{N}^#$ is a hyper inductive if the following statement holds

$$ \bigwedge_{\alpha \in \mathbb{N}^#} (\alpha \in_s S \Rightarrow_s \alpha^+ \in_s S). \qquad (3.7) $$

Obviously a set $\mathbb{N}^#$ is a hyper inductive. Thus axiom of hyper infinite induction 1 asserts that a set $\mathbb{N}^#$ this is the smallest hyper inductive set.

**Axioms of hyperfinite induction 2**

Let $F(x)$ be a wff of the set theory $\mathbf{NC}_{\infty}^{\#}$ strictly restricted on a set $[0, \beta]$ then

$$ \left[ \forall \beta (\beta \in [0, \beta]) \left[ \bigwedge_{0 \le \alpha < \beta} (F(\alpha) \Rightarrow_s F(\alpha^+)) \right] \right] \Rightarrow_s \forall \alpha (\alpha \in [0, \beta]) F(\alpha). \qquad (3.8) $$

Let $F(x)$ be a wff of the set theory $\mathbf{NC}_{\infty}^{\#}$ strictly restricted on a set $[0, \beta_{\infty}]$ then

$$ \left[ \forall \beta (\beta \in [0, \beta_{\infty}]) \left[ \bigwedge_{0 \le \alpha < \beta_{\infty}} (F(\alpha) \Rightarrow_s F(\alpha^+)) \right] \right] \Rightarrow_s \forall \alpha (\alpha \in [0, \beta_{\infty}]) F(\alpha). \qquad (3.9) $$

**Axiom of hyper infinite induction 2**

Let $F(x)$ be anrestricted wff of the set theory $\mathbf{NC}_{\infty}^{\#}$ then

$$ \left[ \forall \beta (\beta \in \mathbb{N}^#) \left[ \bigwedge_{0 \le \alpha < \beta} (F(\alpha) \Rightarrow_s F(\alpha^+)) \right] \right] \Rightarrow_s \forall \beta (\beta \in \mathbb{N}^#) F(\beta). \qquad (3.10) $$

**4. Hyper inductive definitions in general.**
A function $f: \mathbb{N}^\# \to A$ whose domain is the set $\mathbb{N}^\#$ is called an hyper infinite sequence and denoted by $\{f_n\}_{n \in \mathbb{N}^\#}$ or by $\{f(n)\}_{n \in \mathbb{N}^\#}$. The set of all hyper infinite sequences whose terms belong to $A$ is clearly $A^{\mathbb{N}^\#}$; the set of all hyperfinite sequences of $n \in \mathbb{N}^\# \mathbb{N}$ terms in $A$ is $A^n$. The set of all hyperfinite sequences with terms in $A$ can be defined as

$$ \{R \subset \mathbb{N}^\# \times A : (R \text{ is a function}) \wedge \bigvee_{n \in \mathbb{N}^\#} (\mathrm{Dom}(R) = n)\}, \quad (4.1) $$

where $\mathrm{Dom}(R)$ is domain of $R$. This definition implies the existence of the set of all hyper infinite sequences with terms in $A$. The simplest case is the hyper inductive definition of a hyper infinite sequence $\{\varphi(n)\}_{n \in \mathbb{N}^\#}$ (with terms belonging to a certain set $Z$) satisfying the following conditions:

$$ (a) \qquad \varphi(0) = z, \varphi(n^+) = e(\varphi(n), n), \qquad (4.2) $$

where $z \in Z$ and $e$ is a function mapping $Z \times \mathbb{N}^\#$ into $Z$.

More generally, we consider a mapping $f$ of the cartesian product $Z \times \mathbb{N}^\# \times A$ into $Z$ and seek a function $\varphi \in Z^{\mathbb{N}^\# \times A}$ satisfying the conditions:

$$ (b) \qquad \varphi(0,a) = g(a), \varphi(n^+,a) = f(\varphi(n,a),n,a), \qquad (4.3) $$

where $g \in Z^A$. This is a definition by hyper infinite induction with parameter $a$ ranging over the set $A$. Schemes (a) and (b) correspond to induction "from $n$ to $n^+ = n+1$", i.e. $\varphi(n^+)$ or $\varphi(n^+,a)$ depends upon $\varphi(n)$ or $\varphi(n,a)$ respectively. More generally, $\varphi(n^+)$ may depend upon all values $\varphi(m)$ where $m \le n$ (i.e. $m \in n^+$). In the case of induction with parameter, $\varphi(n^+,a)$ may depend upon all values $\varphi(m,a)$, where $m \le n$; or even upon all values $\varphi(m,a)$, where $m \le n^+$ and $b \in A$. In this way we obtain the following schemes of

definitions by hyper infinite induction:

$$ (c) \qquad \varphi(0) = z, \varphi(n^+) = h(\varphi|_{n^+}, n), $$

$$ (d) \qquad \varphi(0,a) = g(a), \quad \varphi(n^+,a) = H(\varphi|(n^+ \times A),n,a). $$

In the scheme (c), $z \in Z$ and $h \in Z^{C \times \mathbb{N}^\#}$, where $C$ is the set of hyperfinite sequences whose terms belong to $Z$; in the scheme (d), $g \in Z^A$ and $H \in Z^{T \times \mathbb{N}^\# \times A}$, where $T$ is the set of functions whose domains are included in $\mathbb{N}^\# \times A$ and whose values belong to $Z$.

It is clear that the scheme (d) is the most general of all the schemes considered above.

By choice of functions one obtains from (d) any of the schemes (a)-(d). For example, taking the function defined by $H(c,n,a) = f(c(n,a),n,a)$ for $a \in A, n \in \mathbb{N}^\#, c \in Z^{\mathbb{N}^\# \times A}$ as $H$ in (d), one obtains (b). We shall now show that, conversely, the scheme (d) can be obtained from (a). Let $g$ and $H$ be functions belonging to $Z^A$ and $Z^{T \times \mathbb{N}^\# \times A}$ respectively, and let $\varphi$ be a function satisfying (d). We shall show that the sequence $\Psi = \{\Psi_n\}_{n \in \mathbb{N}^\#}$ with $\Psi_n = \varphi|(n^+,A)$ can be defined by (a). Obviously, $\Psi_n \in T$ for every $n \in \mathbb{N}^\#$. The first term of the sequence $\Psi$ is equal to $\varphi|(0^+,A)$, i.e. to the set: $z^* = \{\langle\langle 0,a\rangle, g(a)\rangle|a \in A\}$. The relation between $\Psi_n$, and $\Psi_{n^+}$ is given by the formula: $\Psi_{n^+} = \Psi_n \cup \varphi|(\{n^+\} \times A)$, where the second component is

$$ \{\langle\langle n^+, a\rangle, \varphi(n^+, a)\rangle|a \in A\} = \{\langle n^+, a\rangle, H(\Psi_n, n, a)|a \in A\}. \qquad (4.4) $$

Thus we see that the sequence $\Psi$ can be defined by (a) if we substitute $T$ for $Z, z^*$ for $z$ and let $e(c,n) = c \cup \{\langle n^+,a\rangle, H(c,n,a)|a \in A\}$ for $c \in T$.

Now we shall prove the existence and uniqueness of the function satisfying (a).
This theorem shows that we are entitled to use definitions by induction of the type (a).
According to the remark made above, this will imply the existence of functions
satisfying

the formulas (b), (c), and (d). Since the uniqueness of such functions can be proved in
the same manner as for (a), we shall use in the sequel definitions by induction of any
of

the types (a)-(d).

**Theorem 3.1.** If Z is any set $z \in Z$ and $e \in Z^{Z \times \mathbb{N}^#}$, then there exists exactly one hyper infinite sequence $\varphi$ satisfying formulas (a).

**Proof. Uniqueness.** Suppose that $\{\varphi_1(n)\}_{n \in \mathbb{N}^#}$ and $\{\varphi_2(n)\}_{n \in \mathbb{N}^#}$ satisfy (a) and let

$$ K = \{n | n \in \mathbb{N}^# \wedge \varphi_1(n) = \varphi_2(n)\} \qquad (4.5) $$

Then (a) implies that $K$ is hyperinductive. Hence $\mathbb{N}^# = K$ and therefore $\varphi_1(n) = \varphi_2(n)$.

**Existence.** Let $\Phi(z, n, t)$ be the formula $e(z, n) = t$ and let $\Psi(w, z, F_n)$ be the following formula:

$$ (F_n \text{ is a function}) \wedge (\text{Dom}(F) = n^+) \wedge (F(0) = z) \wedge \bigwedge_{m \in n} \Phi(F_n(m), m, F_n(m^+)). \quad (4.6) $$

In other words, $F$ is a function defined on the set of numbers $\le n \in \mathbb{N}^#$ such that $F(0) = z$ and $F(m^+) = e(F(m), m)$ for all $m < n \in \mathbb{N}^#$.

**Assumption 3.1.** We assume now (but without loss of generality) that predicate $\Psi(w, z, F_n)$ is unrestricted on variable $n \in \mathbb{N}^#$, see Definition 3.3.

We prove by hyper infinite induction that there exists exactly one function $F_n$ such that
$\Psi(n, z, F_n)$.

The proof of uniqueness of this function is similar to that given in the Theorem 3.1.

The existence of $F_n$ can be proved as follows: for $n = 0$ it suffices to take $\{\langle 0, z \rangle\}$ as $F_n$; if $n \in \mathbb{N}^#$ and $F_n$ satisfies $\Psi(n, z, F_n)$, then $F_{n^+} = F_n \cup \{\langle n^+, e(F_n(n), n) \rangle\}$

satisfies the condition $\Psi(n^+, z, F_{n^+})$.

Now, we take as $\varphi$ the set of pairs $\langle n, s \rangle$ such that $n \in \mathbb{N}^#; s \in Z$ and

$$ \exists F [\Psi(n, z, F) \wedge (s = F(n))]. \qquad (4.7) $$

Since $F$ is the unique function satisfying $\Psi(n, z, F)$, it follows that $\varphi$ is a function.

For $n = 0$ we have $\varphi(0) = F_0(0) = z$; if $n \in \mathbb{N}^#$, then $\varphi(n^+) = F_{n^+}(n^+) = e(F_n(n), n)$ by the definition of $F_n$; hence we obtain $\varphi(n^+) = e(\varphi(0), n)$. Theorem 3.2 is thus proved.

**Remark 3.1.** Note that Assumption 3.1 is not necessarily, see Appendix B.

We frequently define not one but several functions (with the same range Z) by a
simultaneous induction:

$$ \varphi(0) = z, \varphi(n^+) = f(\varphi(n), \psi(n), n), \psi(0) = t, \psi(n^+) = g(\varphi(n), \psi(n), n) \quad (4.8) $$

where $z, t \in Z$ and $f, g \in Z^{Z \times Z \times \mathbb{N}^#}$.

This kind of definition can be reduced to the previous one. It suffices to notice that the
hyper infinite sequence $\vartheta_n = \langle \varphi(n), \psi(n) \rangle$ satisfies the formulas:

$$ \vartheta_0 = \langle z, t \rangle, \vartheta_{n^+} = e(\vartheta_n, n), \qquad (4.9) $$

where we set

$$ e(u, n) = \langle f(K(u), L(u), n), g(K(u), F(w), n) \rangle, \qquad (4.10) $$

and $K, L$ denote functions such that $K(\langle x, y \rangle)$ and $L(\langle x, y \rangle) = y$ respectively. Thus the function $\vartheta$ is defined by hyper infinite induction by means of (a). We now define $\varphi$ and
$\psi$ by

$$ \varphi(n) = K(\vartheta_n), \psi(n) = L(\vartheta_n). \quad (4.11) $$

**Remark 3.2.** We assume now that predicate $\Psi(w, z, F_n)$ is restricted on variable $n \in \mathbb{N}^\#$, on a set $[0, \beta] \cup \hat{\omega} \subset \mathbb{N}^\#$, see Definition 3.2, then there exists exactly one hyperfinite sequence $\varphi$ satisfying formulas (a). Note that is a case if and only if $f, g \in Z^{Z \times Z \times [0, \beta] \cup \hat{\omega}}$.

# 5. Fundamental examples of the hyper inductive definitions.

## 1. Addition operation of hypernatural numbers

The function $+(m, n) \triangleq m + n : \mathbb{N}^\# \times \mathbb{N}^\# \to \mathbb{N}^\#$ is defined by
$$ m + 0 = m, m + n^+ = (m + n)^+. $$

This definition is obtained from (b) by setting $Z = A = \mathbb{N}^\#$, $g(a) = a, f(p, n, a) = p^+$.

This function satisfies all properties of addition such as: for all $m, n, k \in \mathbb{N}^\#$

(i) $m + 0 = m$ (ii) $m + n = n + m$ (iii) $m + (n + k) = (m + n) + k$.

## 2. Multiplication operation of hypernatural numbers

The function $\times(m, n) \triangleq m \times n : \mathbb{N}^\# \times \mathbb{N}^\# \to \mathbb{N}^\#$ is defined by
$$ m \times 1 = 1, m \times n^+ = m \times n + m. $$

(i) $m \times 1 = 1$ (ii) $m \times n = n \times m$ (iii) $m \times (n \times k) = (m \times n) \times k$.

## 4. Distributivity with respect to multiplication over addition.

$$ m \times (n + k) = m \times n + m \times k. $$

## 5. Let $Z = A = X^X, g(a) = I_X, f(u, n, a) = u \circ a$ in (b). Then (b) takes on the following form

$$ \varphi(0, a) = I_X, \varphi(n^+, a) = \varphi(n, a) \circ a. \quad (5.1) $$

The function $\varphi(n, a)$ is denoted by $a^n$ and is colled $n$-th iteration of the function $a$:

$$ a^0(x) = x, a^{n^+}(x) = a^n(a(x)), x \in X, a \in X^X, n \in \mathbb{N}^\#. \quad (5.2) $$

## 6. Let $A = (\mathbb{N}^\#)^{\mathbb{N}^\#}, g(a) = a_0, f(u, n, a) = u + a_{n^+}$. Then (b) takes on the following form

$$ \varphi(0, a) = a_0, \varphi(n^+, a) = \varphi(n, a) + a_{n^+} \quad (5.3) $$

The function is defined by the Eqs.(5.3) is denoted by

$$ \sum_{i=0}^{n} a_i \quad (5.4) $$

## 7. Let $A = (\mathbb{N}^\#)^{\mathbb{N}^\#}, g(a) = a_0, f(u, n, a) = u \times a_{n^+}$. Then (b) takes on the following form

$$ \varphi(0, a) = a_0, \varphi(n^+, a) = \varphi(n, a) \times a_{n^+} \quad (5.5) $$

The function is defined by the Eqs.(5.5) is denoted by

$$ \prod_{i=0}^{n} a_i \quad (5.6) $$

**Theorem 5.1.** The following equalities holds for any $n, k_1, l_1 \in \mathbb{N}^\#$:

(1) using distributivity

$$ b \times \sum_{i=0}^{n} a_i = \sum_{i=0}^{n} b \times a_i \quad (5.7) $$

(2) using commutativity and associativity

$$ \sum_{i=0}^{n} a_i \pm \sum_{i=0}^{n} b_i = \sum_{i=0}^{n} (a_i \pm b_i) \quad (5.8) $$

(3) splitting a sum, using associativity
$$ \sum_{i=0}^{n} a_i = \sum_{i=0}^{j} a_i + \sum_{i=j+1}^{n} a_i \qquad (5.9) $$

(4) using commutativity and associativity, again

$$ \sum_{i=k_0}^{k_1} \sum_{j=l_0}^{l_1} a_{ij} = \sum_{j=l_0}^{l_1} \sum_{i=k_0}^{k_1} a_{ij} \qquad (5.10) $$

(5) using distributivity

$$ \left( \sum_{i=0}^{n} a_i \right) \times \left( \sum_{j=0}^{n} b_j \right) = \sum_{i=0}^{n} \sum_{j=0}^{n} a_i \times b_j \qquad (5.11) $$

(6)

$$ \left( \prod_{i=0}^{n} a_i \right) \times \left( \prod_{i=0}^{n} b_i \right) = \prod_{i=0}^{n} a_i \times b_i \qquad (5.12) $$

(7)

$$ \left( \prod_{i=0}^{n} a_i \right)^m = \prod_{i=0}^{n} a_i^m \qquad (5.13) $$

**Proof.** Imediately by hyper infinite induction principle.

# 6. Nonstandard arithmetic $A^#.

## Addition Operation of Hypernatural Numbers $N^#

There is a unique binary operation $+(\cdot, \cdot) : N^# \times N^# \rightarrow N^#$ such that

$$ (1) \quad +(m, 0) = m \text{ for all } m \in N^# $$

$$ (2) \quad +(m, n+1) = +(m, n) + 1 \text{ for all } m, n \in N^# $$

This definition satisfies all properties of addition such as

$$ (i) \quad m + 0 = a; (ii) \quad m + n = n + m; (iii) \quad m + (n + k) = (m + n) + k $$

## Multiplication Operation of Hypernatural Numbers $N^#

There is a unique binary operation $\times(m, n) \triangleq m \times n : N^# \times N^# \rightarrow N^#$ such that

$$ m \times 1 = 1, m \times (n + 1) = m \times n + m. $$

$$ (i) \quad m \times 1 = 1 \quad (ii) \quad m \times n = n \times m \quad (iii) \quad m \times (n \times k) = (m \times n) \times k. $$

## Inequalities

The usual total order binary relation $\leq$ on hypernatural numbers $N^#$ defined as follows, assuming 0 is a hypernatural number:

For all $a, b \in N^#$, $a \leq b$ if and only if there exists some $c \in N^#$ such that $a + c = b$.

This relation is stable under addition and multiplication: for $a, b, c \in N^#$, if $a \leq b$, then: $a + c \leq b + c$, and $a \times c \leq b \times c$.

**Proposition 6.1.** (a) For any natural or hypernatural number $k \in N^#$,

$$ \vdash \bigvee_{0 \leq m \leq k} (x = m) \iff s \leq k. \qquad (6.1) $$

(a') For any natural or hypernatural number $\kappa \in N^#$ and any wff $B(x)$ unbounded on variable $x$
$$ \vdash \bigwedge_{0 \le m \le k} B(m) \iff_s \forall x (x \le k \Rightarrow_s B(x)), \qquad (6.2) $$

i.e.

$$ \vdash \forall k (k \in \mathbb{N}^#) \left[ \bigwedge_{0 \le m \le k} B(m) \right] \iff_s \forall k (k \in \mathbb{N}^#) \forall x (x \le k \Rightarrow_s B(x)). \qquad (6.3) $$

(b) For any natural or hypernatural number $k \in \mathbb{N}^#$ such that $k > 0$,

$$ \vdash \bigvee_{1 \le m \le k} (x = m - 1) \iff_s x < k. \qquad (6.4) $$

(b') For any natural or hypernatural number $k \in \mathbb{N}^#$ such that $k > 0$ and any wff $B(x)$ unbounded on variable $x$

$$ \vdash \bigwedge_{0 \le m \le k-1} B(m) \iff_s \forall x (x < k \Rightarrow_s B(x)). \qquad (6.5) $$

(c) For any wff $B(x)$ strictly restricted on a set $[0,y)$ and any wff $E(x)$ strictly restricted on a set $\mathbb{N}^\#\setminus[0,y)$

$$ \vdash (\forall x (x < y \Rightarrow_s B(x))) \wedge (\forall x (x \ge y \Rightarrow_s E(x))) \Rightarrow_s \forall x (B(x) \lor E(x)). \qquad (6.6) $$

**Proof.** (a) We prove $\bigvee_{0 \le m \le k} (x = m) \iff_s x \le k$ by hyperfinite induction in the

metalanguage on $k$. The case for $k = 0, \vdash x = 0 \iff_s x \le 0$, is obvious from the definitions. Assume as inductive hypothesis that

$$ \bigvee_{0 \le m \le k} (x = m) \iff_s x \le k. \qquad (6.7) $$

Now assume that

$$ \left[ \bigvee_{0 \le m \le k} (x = m) \right] \vee (x = k + 1). \qquad (6.8) $$

But $\vdash x = k + 1 \Rightarrow_s x \le k + 1$ and, by the inductive hypothesis,

$$ \bigvee_{0 \le m \le k} (x = m). \qquad (6.9) $$

Also $\vdash x \le k \Rightarrow_s x < k + 1$. Thus, $x \le k + 1$. So,

$$ \vdash \bigvee_{0 \le m \le k+1} (x = m) \Rightarrow_s x \le k + 1. \qquad (6.10) $$

Conversely, assume $x \le k + 1$. Then $x = k + 1 \lor x < k + 1$. If $x = k + 1$, then

$$ \bigvee_{0 \le m \le k+1} (x = m). \qquad (6.11) $$

If $x < k + 1$, then we have $x \le k$. By the inductive hypothesis,

$$ \bigvee_{0 \le m \le k} (x = m). \qquad (6.12) $$

and, therefore,

$$ \bigvee_{0 \le m \le k+1} (x = m). \qquad (6.13) $$

Thus in either case,
$$ \bigvee_{0 \le m \le k+1} (x = m). \qquad (6.14) $$

This proves

$$ \vdash x \le k+1 \Rightarrow_s \bigvee_{0 \le m \le k+1} (x = m). \qquad (6.15) $$

From the inductive hypothesis, we have derived

$$ \bigvee_{0 \le m \le k+1} (x = m) \iff_s x \le k+1 \qquad (6.16) $$

and this completes the proof to part (a). Parts (a'), (b), and (b') follow easily from part (a). Part (c) follows almost immediately from the statement

$t \neq r \Rightarrow_s (t < r) \lor (r < t)$, using obvious tautologies.

**Proposition 6.2.** (a) For any natural or hypernatural number $k \in \mathbb{N}^\#$ and any wff $B(x)$ unbounded on variable $x$

$$ \vdash \forall k \in \mathbb{N}^\#) \forall m (0 \le m \le k) B(m) \iff_s \forall k \in \mathbb{N}^\#) \forall x (x \le k \Rightarrow_s B(x)). \qquad (6.17) $$

**Proof.** It follows from (a') and definition of hyper infinite conjunction

$\bigwedge_{0 \le m \le k} B(m)$, see Appendix A.

**Proposition 6.3.** (a) Axiom of hyperfinite induction (3.5) can be expressed by usual set theoretical language in the following form

$$ \forall S(\subset_s [0, \beta]) \forall \beta(\beta \in \mathbb{N}^\#) \{\forall \alpha(0 \le \alpha < \beta)[(\alpha \in S \Rightarrow \alpha^+ \in S)] \Rightarrow_s S = [0, \beta]\}. \qquad (6.18) $$

(b) Axiom of hyper infinite induction (3.6) can be expressed by usual set theoretical language in the following form

$$ \forall S(S \subset_s \mathbb{N}^\#) \{\forall \beta(\beta \in \mathbb{N}^\#) \forall \alpha(0 \le \alpha < \beta)[(\alpha \in S \Rightarrow_s \alpha^+ \in S)] \Rightarrow_s S = \mathbb{N}^\#\}. \qquad (6.19) $$

**Proof.** (a)-(b) It follows from Proposition 6.3 and definition of hyper infinite conjunction, see Appendix A.

There are several stronger forms of the hyper infinite induction principles that we can prove at this point.

**Theorem 6.1.** (Complete hyperfinite induction 1) (i) Let $S \subset [0, \beta]$, then

$$ \forall x(x \in [0, \beta]) [\forall z(z < x \Rightarrow_s z \in_s S) \Rightarrow x \in_s S] \Rightarrow_s S = [0, \beta] \qquad (6.20) $$

(ii) Let $S \subset [0, \beta_\infty]$, then

$$ \forall x(x \in [0, \beta_\infty]) [\forall z(z < x \Rightarrow_s z \in_s S) \Rightarrow x \in_s S] \Rightarrow_s S = [0, \beta_\infty] \qquad (6.21) $$

**Theorem 6.2.** (Complete hyperfinite induction 2) (i) Let $B(x)$ be a wff of the set theory $\mathbb{NC}^\#_\infty$ strictly restricted on a set $[0, \beta] \subset \mathbb{N}^\# \setminus \mathbb{N}$, then

$$ \forall x(x \in [0, \beta]) [\forall z(z < x \Rightarrow_s B(z)) \Rightarrow_s B(x)] \Rightarrow_s \forall x(x \in [0, \beta]) B(x) \qquad (6.22) $$

(ii) Let $F(x)$ be a wff of the set theory $\mathbb{NC}^\#_\infty$ strictly restricted on a set $[0, \beta_\infty]$, then

$$ \forall x(x \in [0, \beta_\infty]) [\forall z(z < x \Rightarrow_s B(z)) \Rightarrow_s B(x)] \Rightarrow_s \forall x(x \in [0, \beta_\infty]) B(x) \qquad (6.23) $$

**Theorem 6.3.** (Complete hyper infinite induction 2) Let $B(x)$ be anrestricted wff of the set theory $\mathbb{NC}^\#_\infty$ then

$$ \forall x(x \in \mathbb{N}^\#) [\forall z(z < x \Rightarrow_s B(z)) \Rightarrow_s B(x)] \Rightarrow_s \forall x(x \in \mathbb{N}^\#) B(x) \qquad (6.24) $$

In ordinary languagel consider a property $B(x)$ such that, for any $x$, if $B(x)$ holds for all hypernatural numbers less than $x$, then $B(x)$ holds for $x$ also. Then $B(x)$ holds for all
natural and hypernatural numbers $x \in \mathbb{N}^#.

**Proof.** Let $E(x)$ be a wff $\forall z(z \le x \Rightarrow_s B(z))$.

(i) 1. Assume that $\forall x(x \in \mathbb{N}^#)[\forall z(z < x \Rightarrow_s B(z)) \Rightarrow_s B(x)]$, then

2. $[\forall z(z < 0 \Rightarrow_s B(z)) \Rightarrow_s B(0)]$ it follows from 1.

3. $z \prec 0$, then

4. $\forall z(z < 0 \Rightarrow_s B(z))$ it follows from 1,

5. $B(0)$ it follows from 2,4 by MP

6. $\forall z(z \le 0 \Rightarrow_s B(z))$ i.e., $E(0)$ holds it follows from Proposition 6.1(a')

7. $\forall x(x \in \mathbb{N}^#)[\forall z(z < x \Rightarrow_s B(z)) \Rightarrow_s B(x)] \vdash E(0)$ it follows from 1,6 by MP

(ii) 1. Assume that: $\forall x(x \in \mathbb{N}^#)[\forall z(z < x \Rightarrow_s B(z)) \Rightarrow_s B(x)]$.

2. Assume that: $E(x) \equiv \forall z(z \le x \Rightarrow_s B(z))$, then

3. $\forall z(z < x^+ \Rightarrow_s B(z))$ it follows from 2 since $z \le x \Rightarrow z < x^+$.

4. $\forall x(x \in \mathbb{N}^#)[\forall z(z < x^+ \Rightarrow_s B(z)) \Rightarrow_s B(x^+)]$ it follows from 1 by rule A4:if $t$ is free for $x$ in $B(x)$, then $\forall x B(x) \vdash B(t)$.

5. $B(x^+)$ it follows from 3,4 by MP rule.

6. $z \le x^+ \Rightarrow_s z < x^+ \lor z = x^+$ it follows from definitions.

7. $z < x^+ \Rightarrow_s B(z)$ it follows from 3 by particularization rule (PR), see Appendix A.

8. $z = x^+ \Rightarrow_s B(z)$ it follows from 5.

9. $E(x^+) \equiv \forall z(z \le x^+ \Rightarrow_s B(z))$ it follows from 6,7,8,rule Gen.

10. $\forall x(x \in \mathbb{N}^#)[\forall z(z < x \Rightarrow_s B(z)) \Rightarrow_s B(x)] \vdash \forall x(x \in \mathbb{N}^#)[E(x) \Rightarrow_s E(x^+)]$ it follows from 1,9 by generalized deduction theorem,rule Gen.

Now by (i), (ii) and the induction axiom, we obtain $D \vdash \forall x(x \in \mathbb{N}^#)E(x)$ that is
$D \vdash \forall x(x \in \mathbb{N}^#)[\forall z(z < x \Rightarrow B(z))],$ where

$D \equiv \forall x(x \in \mathbb{N}^#)[\forall z(z < x \Rightarrow_s B(z)) \Rightarrow_s B(x)]$.

Hence, by rule A4 twice, $D \vdash x \le x \Rightarrow_s B(x)$. But $x \le x$. So, $D \vdash B(x)$, and, by Gen and the generalized deduction theorem (see Appendix A),

$D \vdash \forall x(x \in \mathbb{N}^#)B(x)$.

Remind that.

**Theorem.** (Complete $\omega$-induction)

$$ \forall S(S \subset \mathbb{N}) \forall x[\forall z(z < x \Rightarrow_s z \in_S S) \Rightarrow x \in_S S] \Rightarrow_S S = \mathbb{N}. \qquad (6.25) $$

**Theorem.** ($\mathbb{N},<$) is a well-ordered set.

**Proof.** We will prove by reductio ad absurdum using complete $\omega$-induction (6.25).

Let $X$ be a nonempty subset of $\mathbb{N}$. Suppose $X$ does not have a $<$-least element.
Then consider the set $\Delta = \mathbb{N}X$.

**Case 1.** $\mathbb{N}X = \emptyset$. Then $X = \mathbb{N}$ and so 0 is a $<$-least element. Contradiction.

**Case 2.** $\mathbb{N}X \neq \emptyset$. There exists an $n \in \mathbb{N}X$ such that for all $k < n, k \in \mathbb{N}X$.

Note that such $n$ necessarily exists because $0 \in \mathbb{N}X$, else $0 \in X$ and would be a $<$-least element of $X$.

Since we have supposed that $\Delta = \mathbb{N}X$ does not have a $<$-least element, thus $n \notin X$. Thus we see that for all $k < n, k \in \Delta \Rightarrow_s n \in \mathbb{N}X$, i.e. a set $\Delta$ has a property

$$ \forall n[\forall k(k < n \Rightarrow_s k \in_s \Delta) \Rightarrow n \in_s \Delta] \qquad (6.26) $$

Using complete $\omega$-induction (6.25) we can conclude that $n \in \mathbb{N}X$ for all $n \in \mathbb{N}$.

Thus $\mathbb{N}X = \mathbb{N}$. But $\mathbb{N}X = \mathbb{N}$ implies $X = \emptyset$. This is a contradiction to $X$ being a nonempty subset of $\mathbb{N}$.
**Remark 6.1.** Let $X$ be a nonempty subset of $\mathbb{N}^#$. Suppose $X$ does not have a $<$-least element. Then consider the set $\Delta^# = \mathbb{N}^#\X \neq \emptyset$. There exists an $n \in \mathbb{N}^#\X$ such that for all $k < n, k \in \mathbb{N}^#\X$. Note that such $n$ necessarily exists because $0 \in \mathbb{N}^#\X$, else $0 \in X$ and would be a $<$-least element of $X$. Since we have supposed that $\Delta^# = \mathbb{N}^#\X$ does not have a least element, thus $n \notin X$.

Therefore we see that for all $k < n, k \in \Delta^# \Rightarrow_s n \in \mathbb{N}^#\X$, i.e. a set $\Delta^#$ has a property

$$ \forall n[\forall k(k < n \Rightarrow_s k \in_s \Delta^#) \Rightarrow n \in_s \Delta^#]. \qquad (6.27) $$

But in contrast with (6.26) we can not conclude from (6.27) that $n \in \mathbb{N}^#\X$ for all $n \in \mathbb{N}^#$. For example let $X$ be a set $\mathbb{N}^#\N$. Thus $\Delta^# = \mathbb{N}$ and (6.27) is satisfied but $\mathbb{N} \neq \mathbb{N}^#$. Obviously $\mathbb{N}^#\N$ does not have a $<$-least element.

**Definition 6.1.** A sequence $\{u_n\}_{n \in \mathbb{Z}}$, $u_n \in \mathbb{N}^#\N$ is a blok corresponding to a gyperfinite number $u = u_0 \in \mathbb{N}^#\N$ iff there is a gyperfinite number $u$ such that $\dots \in u_{-(n+1)} \in u_{-n} \dots \in u_{-4} \in u_{-3} \in u_{-2} \in u_{-1} \in u$ and the following conditions are satisfied

$$ \dots \in u_{-(n+1)} \in u_{-n} \dots \in u_{-4} \in u_{-3} \in u_{-2} \in u_{-1} \in u \in u_1 \in u_2 \dots \in u_n \in u_{n+1} \in \dots \quad (6.28) $$

where for any $n \in \mathbb{N}$: $u_{-(n+1)} \in u_{-n}$, where $u_{-n} = u^{+(n+1)}$.

Thus beginning with an infinite integer $u \in \mathbb{N}^#\N$ we obtain a block (8.20) of infinite integers. However, given a "block," there is another block consisting of even larger infinite integers. For example, there is the integer $u+u$, where $u+k < u+u$ for each $k \in \mathbb{N}$. And $v = u+u$ is itself part of the block:

$$ \dots < v - 3 < v - 2 < v - 1 < v < v + 1 < v - 2 < \dots \qquad (6.29) $$

Of course, $v < v+u < v+v$, and so forth. There are even infinite integers $u \times u$ and $u^u$, and so forth. Proceeding in the opposite direction, if $u \in \mathbb{N}^#\N$, either $u$ or $u+1$ is of the form $v+v$. Here $v$ must be infinite. So there is no first block, since $v < u$. In fact, the ordering of the blocks is dense. For let the block containing $v$ precede the one containing $u$, that is,

$$ v - 2 < v - 1 < v < v + 1 < \dots < u - 2 < u - 1 < u < u + 1 < \dots \qquad (6.30) $$

Either $u+v$ or $u+v+1$ can be written $z+z$ where $v+k < z < u-l$ for all $k,l \in \mathbb{N}$.

**Remark 6.2.** Note that $\mathbb{N}^#$ consists of $\mathbb{N}$ as an initial segment followed by an ordered set of blocks. These blocks are densely ordered with no first or last element. Each block is itself order-isomorphic to the integers

$$ \dots -3, -2, -1, 0, 1, 2, 3, \dots \qquad (6.31) $$

Although $\mathbb{N}^#\N$ is a nonempty subset of $\mathbb{N}^#$, as we have just seen it has no least element and likewise for any block.

# 7. Hyperrationalem Q#.

Now that we have the hypernatural numbers $\mathbb{N}^#$, defining hyperintegers and hyperrational numbers is well within reach [2].

**Definition 7.1.** Let $Z^{\prime} = \mathbb{N}^# \times \mathbb{N}^#$. We can define an equivalence relation $\approx$ on $Z^{\prime\prime}$ by $(a,b) \approx (c,d)$ if and only if $a+d = b+c$. Then we denote the set of all hyperintegers by $Z^{\prime} = Z^{\prime\prime}/\approx$ (The set of all equivalence classes of $Z^{\prime\prime}$ modulo $\approx$).

**Definition 7.2.** Let $Q^{\prime\prime} = Z^{\prime} \times (Z^{\prime} - \{0\}) = \{(a,b) \in Z^{\prime} \times Z^{\prime}|b \neq 0\}$. We can define an equivalence relation $\approx$ on $Q^{\prime\prime}$ by $(a,b) \approx (c,d)$ if and only if $a \times d = b \times c$. Then we
denote

the set of all hyperrational numbers by $\mathbb{Q}^# = Q'/\approx$ (The set of all equivalence classes of $Q'$ modulo $\approx$).

**Definition 7.3.** A linearly ordered set $(P, <)$ is called dense if for any $a, b \in P$ such that $a < b$, there exists $z \in P$ such that $a < z < b$.

**Lemma 7.1.** $(\mathbb{Q}^#<\emptyset)$ is dense.

**Proof.** Let $x = (a,b), y = (c,d) \in \mathbb{Q}^#$ be such that $x < y$. Consider $z = (ad+bc, 2bd) \in \mathbb{Q}^#$.

It is easily shown that $x < z < y$.

## 8. External Cauchy hyperreals $\mathbb{R}_c^#$ via Cauchy completion.

**Definition 8.1.** A hyper infinite sequence of hyperrational numbers (or for the sake of brevity simply hyperrational sequence) is a function from the hypernatural numbers $\mathbb{N}^#$ into the hyperrational numbers $\mathbb{Q}^#$. We usually denote such a function by $n \mapsto a_n$, or by $a : n \to a_n$, so the terms in the sequence are written $\{a_1, a_2, a_3, \dots, a_n\dots\}$. To refer to the whole hyper infinite sequence, we will write $\{a_n\}_{n=1}^{\infty}$, or $\{a_n\}_{n \in \mathbb{N}^#}$, or for the sake of brevity simply $\{a_n\}$.

**Definition 8.2.** Let $\{a_n\}$ be a hyperrational sequence. Say that $\{a_n\}$ #-tends to 0 if, given any $\varepsilon > 0$, $\varepsilon \approx 0$, there is a hypernatural number $N \in \mathbb{N}^#\mathbb{N}$, $N = N(\varepsilon)$ such that, after $N$ (i.e. for all $n > N$), $|a_n|\le \varepsilon$. We often denote this symbolically by $a_n \to_\# 0$. We can also, at this point, define what it means for a hyperrational sequence #-tends to any given number $q \in \mathbb{Q}^#$: $\{a_n\}$ #-tends to $q$ if the hyperrational sequence $\{a_n - q\}$ #-tends to 0 i.e., $a_n - q \to_\# 0$.

**Definition 8.3.** Let $\{a_n\}$ be a hyper infinite hyperrational sequence. We call $\{a_n\}$ a Cauchy hyperrational sequence if the difference between its terms #-tends to 0. To be precise: given any hyperrational number $\varepsilon > 0$, $\varepsilon \approx 0$, there is a hypernatural number $N = N(\varepsilon)$ such that for any $m, n > N$, $|a_n - a_m| < \varepsilon$.

**Theorem 8.1.** If $\{a_n\}$ is a #-convergent hyperrational sequence (that is, $a_n \to_\# q$ for some hyperrational number $q \in \mathbb{Q}^#$), then $\{a_n\}$ is a Cauchy hyperrational sequence.

**Proof.** We know that $a_n \to_\# q$. Here is a ubiquitous trick: instead of using $\varepsilon$ in the definition, start with an arbitrary small $\varepsilon > 0$, $\varepsilon \approx 0$ and then choose $N \in \mathbb{N}^#\mathbb{N}$ so that $|a_n - q| < \varepsilon/2$ when $n > N$. Then if $m, n > N$, we have $|a_n - a_m| = |(a_n - q) - (a_m - q)| \le |a_n - q| + |a_m - q| < \varepsilon/2 + \varepsilon/2 = \varepsilon$. This shows that $\{a_n\}$ is a Cauchy hyper infinite sequence.

**Theorem 8.2.** If $\{a_n\}$ is a Cauchy hyperrational sequence, then it is bounded or hyper bounded; that is, there is some $M \in \mathbb{Q}^#$ finite or hyperfinite such that $|a_n| \le M$ for all $n \in \mathbb{N}^#$.

**Proof.** Since $\{a_n\}$ is Cauchy, setting $\varepsilon = 1$ we know that there is some $N \in \mathbb{N}^#\mathbb{N}$ such that $|a_m - a_n| < 1$ whenever $m, n > N$. Thus, $|a_{N+1} - a_n| < 1$ for $n > N$. We can rewrite this as $a_{N+1} - 1 < a_n < a_{N+1} + 1$. This means that $|an|$ is less than the maximum of $|a_{N+1} - 1|$ and $|a_{N+1} + 1|$. So, set $M$ equal to the maximum number in the following list: $\{|a_0|, |a_1|, \dots, |a_N|, |a_{N+1} - 1|, |a_{N+1} + 1|\}$. Then for any term $a_n$, if $n \le N$, then $|a_n|$ appears in the list and so $|a_n| \le M$; if $n > N$, then (as shown above) $|a_n|$ is less than at least one of the last two entries in the list, and so $|a_n| \le M$. Hence, $M$ is a bound for the sequence.
**Definition 8.4.** Let $S$ be a set. A relation $x \sim y$ among pairs of elements of $S$
is said to be an equivalence relation if the following three properties hold:

Reflexivity: for any $s \in S, s \sim s$.

Symmetry: for any $s, t \in S$, if $s \sim t$ then $t \sim s$.

Transitivity: for any $s, t, r \in S$, if $s \sim t$ and $t \sim r$, then $s \sim r$.

**Theorem 8.3.** Let $S$ be a set, with an equivalence relation $(\cdot, \sim)$ on pairs of elements.
For $s \in S$, denote by $cl[s]$ the set of all elements in $S$ that are related to $s$. Then for

any $s, t \in S$, either $cl[s] = cl[t]$ or $cl[s]$ and $cl[t]$ are disjoint.

The hyperreal numbers $\mathbb{R}_c^\#$ will be constructed as equivalence classes of Cauchy
hyperrational sequences. Let $\mathcal{F}_{\mathbb{Q}^#}$ denote the set of all Cauchy hyperrational
sequences of hyperrational numbers. We define the equivalence relation on $\mathcal{F}_{\mathbb{Q}^#}$.

**Definition 8.5.** Let $\{a_n\}$ and $\{b_n\}$ be in $\mathcal{F}_{\mathbb{Q}^#}$. Say they are #-equivalent if $a_n - b_n \to_{\#} 0$ i.e., if and only if the hyperrational sequence $\{a_n - b_n\}$ tends to 0.

**Theorem 8.4.** Definition 8.4 yields an equivalence relation on $\mathcal{F}_{\mathbb{Q}^#}$.

**Proof.** We need to show that this relation is reflexive, symmetric, and transitive.

**Reflexive:** $a_n - a_n = 0$, and the sequence all of whose terms are 0 clearly 
\#-converges to 0. So $\{a_n\}$ is related to $\{a_n\}$.

**Symmetric:** Suppose $\{a_n\}$ is related to $\{b_n\}$, so $a_n - b_n \to_{\#} 0$.

But $b_n - a_n = -(a_n - b_n)$, and since only the absolute value $|a_n - b_n| = |b_n - a_n|$ comes into play in Definition 8.2, it follows that $b_n - a_n \to_{\#} 0$ as well. Hence, $\{b_n\}$ is related to $\{a_n\}$.

**Transitive:** Here we will use the $\varepsilon/2$ trick we applied to prove Theorem 3.1. Suppose $\{a_n\}$ is related to $\{b_n\}$, and $\{b_n\}$ is related to $\{c_n\}$. This means that $a_n - b_n \to_{\#} 0$ and $b_n - c_n \to_{\#} 0$. To be fully precise, let us fix $\varepsilon > 0$, $\varepsilon \approx 0$; then there exists an $N \in \mathbb{N}^\#$ such that for all $n > N$, $|a_n - b_n| < \varepsilon/2$; also, there exists an $M$ such that for all $n > M$, $|b_n - c_n| < \varepsilon/2$. Well, then, as long as $n$ is bigger than both $N$ and $M$, we have that $|a_n - c_n| \le |(a_n - b_n) + (b_n - c_n)| \le |a_n - b_n| + |b_n - c_n| < \varepsilon/2 + \varepsilon/2 = \varepsilon$.

So, choosing $L$ equal to the max of $N, M$, we see that given $\varepsilon > 0$ we can always choose $L$ so that for $n > L$, $|a_n - c_n| < \varepsilon$. This means that $a_n - c_n \to_{\#} 0$ i.e. $\{a_n\}$ is related to $\{c_n\}$.

**Definition 8.6.** The hyperreal numbers $\mathbb{R}_c^\#$ are the equivalence classes $cl[\{a_n\}]$ of Cauchy sequences of hyperrational numbers, as per Definition 3.5. That is, each such equivalence class is a hyperreal number.

**Definition 8.7.** Given any hyperrational number $q \in \mathbb{Q}^#$, define a hyperreal number $q^\#$ to be the equivalence class of the sequence $q^\# = (q, q, q, q, \dots)$ consisting entirely of $q$. So we view $\mathbb{Q}^#$ as being inside $\mathbb{R}_c^\#$ by thinking of each hyperrational number $q \in \mathbb{Q}^#$ as its associated equivalence class $q^\#$. It is standard to abuse this notation, and simply refer to the equivalence class as $q$ as well.

**Definition 8.8.** Let $s, t \in \mathbb{R}_c^\#$, so there are Cauchy sequences $\{a_n\}, \{b_n\}$ of hyperrational numbers with $s = cl[\{a_n\}]$ and $t = cl[\{b_n\}]$.

(a) Define $s + t$ to be the equivalence class of the sequence $\{a_n + b_n\}$.

(b) Define $s \times t$ to be the equivalence class of the sequence $\{a_n \times b_n\}$.

**Theorem 8.5.** The operations $+, \times$ in Definition 8.8 (a), (b) are well-defined.

**Proof.** Suppose that $cl[\{a_n\}] = cl[\{c_n\}]$ and $cl[\{b_n\}] = cl[\{d_n\}]$. Thus means that $a_n - c_n \to_{\#} 0$ and $b_n - d_n \to_{\#} 0$. Then $(a_n + b_n) - (c_n + d_n) = (a_n - c_n) + (b_n - d_n)$.

Now, using the familiar $\varepsilon/2$ trick, you can construct a proof that this tends to 0, and
so $\mathrm{cl}[\{a_n + b_n\}] = \mathrm{cl}[\{c_n + d_n\}]$.

Multiplication is a little trickier; this is where we will use Theorem 8.2. We will also use another ubiquitous technique: adding 0 in the form of $s - s$. Again, suppose that $\mathrm{cl}[(a_n) = \mathrm{cl}[(c_n)]$ and $\mathrm{cl}[\{b_n\}] = \mathrm{cl}[\{d_n\}]$; we wish to show that $\mathrm{cl}[\{a_n \times b_n\}] = \mathrm{cl}[\{c_n \times d_n\}]$, or, in other words, that $a_n \times b_n - c_n \cdot d_n \to_\# 0$. Well, we add and subtract one of the other cross terms, say

$$b_n \times c_n : a_n \times b_n - c_n \times d_n = a_n \times b_n + (b_n \times c_n - b_n \times c_n) - c_n \times d_n = \\ = (a_n \times b_n - b_n \times c_n) + (b_n \times c_n - c_n \times d_n) = b_n \times (a_n - c_n) + c_n \times (b_n - d_n).$$

Hence, we have $|a_n \times b_n - c_n \times d_n| \le |b_n| \times |a_n - c_n| + |c_n| \times |b_n - d_n|$. Now, from Theorem 8.2, there are numbers $M$ and $L$ such that $|b_n| \le M$ and $|c_n| \le L$ for all $n \in \mathbb{N}^\#$. Taking some number $K$ which is bigger than both, we have

$$|a_n \times b_n - c_n \times d_n| \le |b_n| \times |a_n - c_n| + |c_n| \times |b_n - d_n| \le K(|a_n - c_n| + |b_n - d_n|).$$

Now, noting that both $a_n - c_n$ and $b_n - d_n$ tend to 0 and using the $\varepsilon/2$ trick (actually, this time we'll want to use $\varepsilon/2K$), we see that $a_n \times b_n - c_n \times d_n \to_\# 0$.

**Theorem 8.6.** Given any hyperreal number $s \ne 0$, there is a hyperreal number $t$ such that $s \times t = 1$.

**Proof.** First we must properly understand what the theorem says. The premise is that $s$ is nonzero, which means that $s$ is not in the equivalence class of $\{0,0,0,0,...\}$. In other words, $s = \mathrm{cl}[\{a_n\}]$ where $a_n \to 0$ does not #-converge to 0. From this, we are to deduce the existence of a hyperreal number $t = \mathrm{cl}[\{b_n\}]$ such that $s \times t = \mathrm{cl}[\{a_n \times b_n\}]$ is the same equivalence class as $\mathrm{cl}[\{1, 1, 1, 1,...\}]$. Doing so is actually an easy consequence of the fact that nonzero rational numbers have multiplicative inverses, but there is a subtle difficulty. Just because $s$ is nonzero (i.e. $\{a_n\}$ does not tend to 0), there's no reason any number of the terms in $\{a_n\}$ can't equal 0. However, it turns out that eventually, $a_n \ne 0$.

That is,

**Lemma 8.1.** If $\{a_n\}$ is a Cauchy hyper infinite sequence which does not #-tend to 0, then there is an $N \in \mathbb{N}^\#/\mathbb{N}$ such that, for $n > N, a_n \ne 0$.

We will now use Lemma 8.1 to complete the proof of Theorem 8.7.

Let $N$ be such that $a_n \ne 0$ for $n > N$. Define a hyper infinite sequence $b_n$ of hyperrational numbers as follows:

for $n \le N, b_n = 0$, and for $n > N, b_n = 1/a_n; \{b_n\} = (0,0,...,0, 1/a_{N+1}, 1/a_{N+2},...)$.

This makes sense since, for $n > N$, an is a nonzero hyperrational number, so $1/a_n$ exists. Then $a_n \cdot b_n$ is equal to $a_n \cdot 0 = 0$ for $n \le N$, and equals $a_n \cdot b_n = a_n \cdot 1/a_n = 1$ for $n > N$. Well, then, if we

look at the hyper infinite sequence $(1, 1, 1, 1,...)$, we have $(1, 1, 1, 1,...) - (a_n \cdot b_n)$ is the

hyper infinite sequence which is $1 - 0 = 1$ for $n \le N$ and equals $1 - 1 = 0$ for $n > N$.

Since this sequence is eventually equal to 0, it #-converges to 0, and so

$$\mathrm{cl}[\{a_n \cdot b_n\}] = \mathrm{cl}[(1, 1, 1, 1, ...)] = 1 \in \mathbb{R}^\#_c. \text{ This shows that } t = \mathrm{cl}[\{b_n\}] \text{ is a} \\ \text{multiplicative inverse to } s = \mathrm{cl}[\{a_n\}].$$

**Definition 8.9.** Let $s \in \mathbb{R}^\#_c$. Say that $s$ is positive if $s \ne 0$, and if $s = \mathrm{cl}[\{a_n\}]$ for some Cauchy sequence of hyperrational numbers such that for some $N \in \mathbb{N}^\#$, $a_n > 0$ for all
$n > N$. Given two hyperreal numbers $s, t$, say that $s > t$ if $s - t$ is positive.

**Theorem 8.7.** Let $s, t$ be hyperreal numbers such that $s > t$, and let $r \in \mathbb{R}_c^#$. Then $s + r > t + r$.

**Proof.** Let $s = \text{cl}[\{a_n\}]$, $t = \text{cl}[\{b_n\}]$, and $r = \text{cl}[\{c_n\}]$. Since $s > t$ i.e., $s - t > 0$, we know that there is an $N \in \mathbb{N}^#$ such that, for $n > N$, $a_n - b_n > 0$. So $a_n > b_n$ for $n > N$. Now, adding $c_n$ to both sides of this inequality (as we know we can do for hyperrational numbers), we have $a_n + c_n > b_n + c_n$ for $n > N$, or $(a_n + c_n) - (b_n + c_n) > 0$ for $n > N$. Note also that $(a_n + c_n) - (b_n + c_n) = a_n - b_n$ does not #-converge to 0, by the assumption that $s - t > 0$. Thus, by Definition 8.8, this means that

$$s + r = \text{cl}[\{a_n + c_n\}] > \text{cl}[\{b_n + c_n\}] = t + r.$$

**Theorem 8.8.** (Generalized Archimedean property) Let $s, t > 0$ be hyperreal numbers. Then there is $m \in \mathbb{N}^#$ such that $m \times s > t$.

**Proof.** Let $s, t > 0$ be hyperreal numbers. We need to find a hypernatural number $m$ so that

$m \times s > t$. First, recall that, by $m$ in this context, we mean $\text{cl}[\{m, m, m, \dots\}]$. So, letting $s = \text{cl}[\{a_n\}]$ and $t = \text{cl}[\{b_n\}]$, what we need to show is that there exists $m$ with $\text{cl}[\{m, m, m, \dots\}] \times \text{cl}[\{a_1, a_2, a_3, a_4, \dots\}] = \text{cl}[\{m \times a_1, m \times a_2, m \times a_3, m \times a_4, \dots\}] > \text{cl}[\{b_1, b_2, b_3, b_4, \dots\}]$.

Now, to say that $\text{cl}[\{m \times a_n\}] > \text{cl}[\{b_n\}]$, or $\text{cl}[\{m \times a_n - b_n\}]$ is positive, is, by Definition 8.9, just to say that there is $N \in \mathbb{N}^#$ such that $m \times a_n - b_n > 0$ for all $n > N$, while $m \times a_n - b_n \not\to_\# 0$. To be precise, the first statement is:
There exist $m, N \in \mathbb{N}^#$ so that $m \times a_n > b_n$ for all $n > N$.

To produce a contradiction, we assume this is not the case; assume that (#) for every $m$ and $N$, there exists an $n > N$ so that $m \times a_n \le b_n$.

Now, since $\{b_n\}$ is a Cauchy sequence, by Theorem 3.2 it is hyperbounded – there is an hyperrational number $M \in \mathbb{Q}^#$ such that $b_n \le M$ for all $n$. Now, by the properties for the hyperrational numbers $\mathbb{Q}^#$, given any hyperrational number $\varepsilon > 0$, $\varepsilon \approx 0$, there is an $m \in \mathbb{N}^#$ such that $M/m < \varepsilon/2$. Fix such an $m$. Then if $m \times a_n \le b_n$, we have $a_n \le b_n/m \le M/m < \varepsilon/2$.

Now, $\{a_n\}$ is a Cauchy sequence, and so there exists $N$ so that for $n, k > N$, $|a_n - a_k| < \varepsilon/2$.

By Asumption (#), we also have an $n > N$ such that $m \times a_n \le b_n$, which means that $a_n < \varepsilon/2$. But then for every $k > N$, we have that $a_k - a_n < \varepsilon/2$, so $a_k < a_n + \varepsilon/2 < \varepsilon/2 + \varepsilon/2 = \varepsilon$. Hence, $a_k < \varepsilon$ for all $k > N$. This proves that $a_k \to_\# 0$, which by Definition 8.9 contradicts the fact that $\text{cl}[\{a_n\}] = s > 0$.

Thus, there is indeed some $m \in N$ so that $m \times a_n - b_n > 0$ for all sufficiently infinite large $n \in \mathbb{N}^# \setminus \mathbb{N}$. To conclude the proof, we must also show that $m \times a_n - b_n \not\to 0$. Actually, it is possible that $m \times a_n - b_n \to 0$ (for example if $\{a_n\} = \{1, 1, 1, \dots\}$ and $\{b_n\} = \{m, m, m, \dots\}$). But that's okay: then we can simply choose a larger $m$. That is: let $m$ be a hypernatural number constructed as above, so that $m \times a_n - b_n > 0$ for all sufficiently large $n \in \mathbb{N}^# \setminus \mathbb{N}$. If it happens to be true that $m \times a_n - b_n \not\to 0$, then the proof is complete.

If, on the other hand, it turned out that $m \times a_n - b_n \to 0$, then take instead the integer $m + 1$. Since $s = \text{cl}[\{a_n\}] > 0$, we have a $n > 0$ for all infinite large $n$, so
$(m+1) \times a_n - b_n = m \times a_n - b_n + a_n > a_n > 0$ for all infinite large $n$, so $m+1$ works just as

well as $m$ did in this regard; and since $m \times a_n - b_n \to 0$, we have

$(m+1) \times a_n - b_n = (m \times a_n - b_n) + a_n \to 0$ since $s = \text{cl}([a_n]) > 0$ (so $a_n \to 0$).

It will be handy to have one more Theorem about how the hyperrationals $\mathbb{Q}^\#$ and hyperreals $\mathbb{R}_c^\#$ compare before we proceed. This theorem is known as the density of $\mathbb{Q}^\#$ in $\mathbb{R}_c^\#$, and it follows almost immediately from the construction of the $\mathbb{R}_c^\#$ from $\mathbb{Q}^\#$.

**Theorem 8.9.** Given any hyperreal number $r \in \mathbb{R}_c^\#$, and any hyperrational number $\varepsilon > 0$, $\varepsilon \approx 0$, there is a hyperrational number $q \in \mathbb{Q}^\#$ such that $|r-q| < \varepsilon$.

**Proof.** The hyperreal number $r$ is represented by a Cauchy hyperrational sequence $\{a_n\}$.

Since this sequence is Cauchy, given $\varepsilon > 0$, $\varepsilon \approx 0$, there is $N \in \mathbb{N}^\#$ so that for all $m, n > N$,

$|a_n - a_m| < \varepsilon$. Picking some fixed $l > N$, we can take the hyperrational number $q$ given by $q = \text{cl}([a_l, a_{l+1}, a_l, \dots])$. Then we have $r-q = \text{cl}([a_n - a_l]_{n \in \mathbb{N}^\#})$, and $q-r = \text{cl}([a_l - a_n]_{n \in \mathbb{N}^\#})$.

Now, since $l > N$, we see that for $n > N$, $a_n - a_l < \varepsilon$ and $a_l - a_n < \varepsilon$, which means by Definition 8.9 that $r-q < \varepsilon$ and $q-r < \varepsilon$; hence, $|r-q| < \varepsilon$.

**Definition 8.10.** Let $S \subsetneq \mathbb{R}_c^\#$ be a non-empty set of hyperreal numbers.

A hyperreal number $x \in \mathbb{R}_c^\#$ is called an upper bound for $S$ if $x \ge s$ for all $s \in S$.

A hyperreal number $x$ is the least upper bound (or supremum) $\sup S$) for $S$ if $x$ is an upper bound for $S$ and $x \le y$ for every upper bound $y$ of $S$.

**Remark 8.1.** The order $\le$ given by Definition 8.9 obviously is $\le$-incomplete.

**Definition 8.11.** Let $S \subsetneq \mathbb{R}_c^\#$ be a nonempty subset of $\mathbb{R}_c^\#$. We will say that:

(1) $S$ is $\le$-admissible above if the following conditions are satisfied:

(i) $S$ bounded above;

(ii) let $A(S)$ be a set $\forall x[x \in A(S) \Leftrightarrow x \ge S]$ then for any $\varepsilon > 0, \varepsilon \approx 0$ there exists $\alpha \in S$ and $\beta \in A(S)$ such that $\beta - \alpha \le \varepsilon \approx 0$.

(2) $S$ is $\le$-admissible below if the following condition are satisfied:

(i) $S$ bounded below;

(ii) let $L(S)$ be a set $\forall x[x \in L(S) \Leftrightarrow x \le S]$ then for any $\varepsilon > 0, \varepsilon \approx 0$ there exists $\alpha \in S$ and $\beta \in L(S)$ such that $\alpha - \beta \le \varepsilon \approx 0$.

**Theorem 8.10.** (i) Any $\le$-admissible above subset $S \subset \mathbb{R}_c^\#$ has the least upper bound property. (ii) Any $\le$-admissible below subset $S \subset \mathbb{R}_c^\#$ has the greatest lower bound property.

**Proof.** Let $S \subset \mathbb{R}_c^\#$ be a nonempty subset, and let $M$ be an upper bound for $S$. We are going to construct two sequences of hyperreal numbers, $\{u_n\}$ and $\{l_n\}$. First, since $S$ is nonempty, there is some element $s_0 \in S$. Now, we go through the following hyperinductive procedure to produce numbers $u_0, u_1, u_2, \dots, u_n, \dots$ and $l_1, l_2, l_3, \dots, l_n, \dots$

(i) Set $u_0 = M$ and $l_0 = s_0$.

(ii) Suppose that we have already defined $u_n$ and $l_n$. Consider the number $m_n = (u_n + l_n)/2$, the average between $u_n$ and $l_n$.

(1) If $m_n$ is an upper bound for $S$, define $u_{n+1} = m_n$ and $l_{n+1} = l_n$.

(2) If $m_n$ is not an upper bound for $S$, define $u_{n+1} = u_n$ and $l_{n+1} = m_n$.
Since $s_0 < M$, it is easy to prove by hyper infinite induction that $\{u_n\}$ is a non-increasing

sequence: $u_{n+1} \le u_n, n \in \mathbb{N}^\#$ and $\{l_n\}$ is a non-decreasing sequence $l_{n+1} \ge l_n, n \in \mathbb{N}^\#$. This

gives us the following lemma.

**Lemma 8.2.** $\{u_n\}$ and $\{l_n\}$ are Cauchy sequences of hyperreal numbers.

**Proof.** Note that each $l_n \le M$ for all $n \in \mathbb{N}^\#$. Since $\{l_n\}$ is non-decreasing, it follows that $\{l_n\}$ is Cauchy. For $\{u_n\}$, we have $u_n \ge s_0$ for all $n \in \mathbb{N}^\#$, and so $-u_n \le -s_0$. Since $\{u_n\}$

is non-increasing, $\{-u_n\}$ is non-decreasing, and so as above, $\{-u_n\}$ is Cauchy. It is easy

to verify that, therefore, $\{u_n\}$ is Cauchy.

The following Lemma shows that $\{u_n\}$ does tend to a hyperreal number.

**Lemma 8.3.** There is a hyperreal number $u$ such that $u_n \to_{\#} u$.

**Proof.** Fix a term $u_n$ in the sequence $\{u_n\}$. By Theorem 8.9, there is a hyperrational number $q_n$ such that $|u_n - q_n| < 1/n$. Consider the sequence $\{q_1, q_2, q_3, \dots, q_n, \dots\}$ of hyperrational numbers. We will show this hypersequence is Cauchy. Fix $\varepsilon > 0, \varepsilon \approx 0$. By the Theorem 8.8, we can choose $N \in \mathbb{N}^\#$ so that $1/N < \varepsilon/3$. We know, since $\{u_n\}$ is Cauchy, that there is an $M \in \mathbb{N}^\#$ such that for $n, m > M, |u_n - u_m| < \varepsilon/3$. Then, so long as $n, m > \max\{N, M\}$, we have

$$|q_n - q_m| = |(q_n - u_n) + (u_n - u_m) + (u_m - q_m)| \leq \\ \leq |q_n - u_n| + |u_n - u_m| + |u_m - q_m| < \varepsilon/3 + \varepsilon/3 + \varepsilon/3 = \varepsilon.$$

Thus, $\{q_n\}$ is a Cauchy sequence of hyperrational numbers, and so it represents a hyperreal number $u = \text{cl}[\{q_n\}]$. We must show that $u_n - u \to_{\#} 0$, but this is practically built into the definition of $u$. To be precise, letting $q_n^*$ be the hyperreal number $\text{cl}[\{q_n, q_n, q_n, \dots\}]$, we see immediately that $q_n^* - u \to_{\#} 0$ (this is precisely equivalent to the statement that $\{q_n\}$ is Cauchy). But $u_n - q_n^* < 1/n$ by construction; it is easily verify that the assertion that if a sequence $q_n^* \to_{\#} u$ and $u_n - q_n^* \to_{\#} 0$, then $u_n \to_{\#} u$. So $\{u_n\}$, a non-increasing sequence of upper bounds for $S$, tends to a hyperreal

number $u$. As you've guessed, $u$ is the least upper bound of our set $S$. To prove this, we

need one more lemma.

**Lemma 8.4.** $l_n \to_{\#} u$.

**Proof.** First, note in the first case above, we have that

$$u_{n+1} - l_{n+1} = m_n - l_n = \frac{u_n + l_n}{2} - l_n = \frac{u_n - l_n}{2}.$$

In the second case, we also have

$$u_{n+1} - l_{n+1} = u_n - m_n = u_n - \frac{u_n + l_n}{2} = \frac{u_n - l_n}{2}.$$

Now, this means that $u_1 - l_1 = \frac{1}{2}(M - s)$, and so $u_2 - l_2 = \frac{1}{2}(u_1 - l_1) = \frac{1}{2^2}(L - s)$, and in general by hyperinfinite induction, $u_n - l_n = 2^{-n}(L - s)$. Since $L > s$ so $L - s > 0$, and since $2^{-n} < 1/n$, by the Theorem 8.8, we have for any $\varepsilon > 0$ that $2^{-n}(L - s) < \varepsilon$ for all sufficiently large $n \in \mathbb{N}^\#/\mathbb{N}$. Thus, $u_n - l_n < 2^{-n}(L - s) < \varepsilon$ as well, and so $u_n - l_n \to_{\#} 0$. Again, it is easily verify that, since $u_n \to_{\#} u$, we have $l_n \to_{\#} u$ as well.
**Proof of Theorem 8.10.** First, we show that $u$ is an upper bound. Well, suppose it is not, so that $u < s$ for some $s \in S$. Then $\varepsilon = s - u$ is $> 0$, and since $u_n \to u$ and is non-increasing, there must be an $n$ so that $u_n - u < \varepsilon$, meaning that $u_n < u + \varepsilon = u + (s - u) = s$. Since $u_n$ is an upper bound for $S$, however, this is a contradiction. Hence, $u$ is an upper bound for $S$.

Now, we also know that, for each $n, l_n$ is not an upper bound, meaning that for each $n$, there is an $s_n \in S$ so that $l_n \le s_n$. Lemma 8.4 tells us that $l_n \to_{\#} u$, and since the sequence $\{l_n\}$ is non-decreasing, this means that for each $\varepsilon > 0$, there is an $N \in \mathbb{N}^\# / \mathbb{N}$ so that for $n > N, l_n > u - \varepsilon$. Hence, for $n > N, s_n \ge l_n > u - \varepsilon$ as well. In particular, for each $\varepsilon > 0$, there is an element $s \in S$ such that $s > u - \varepsilon$. This means that no number smaller than $u$ can be an upper bound for $S$. Hence, $u$ is the least upper bound for $S$.

**Remark 8.2.** Note that assumption in Theorem 8.10 that $S$ is $\le$-admissible above subset of $\mathbb{R}_c^\#$ is necessarily, othervice Theorem 8.10 is not holds. For example let $\Delta = \{\varepsilon|\varepsilon \ge 0 \land \varepsilon \approx 0\}$. Obviously a set $\Delta$ is not $\le$-admissible above subset of $\mathbb{R}_c^\#$. It is clear that Theorem 8.10 is not holds for a set $\Delta$.

**Theorem 8.11. (Generalized Nested Intervals Theorem)**

Let $\{I_n\}_{n \in \mathbb{N}^\#} = \{[a_n, b_n]\}_{n \in \mathbb{N}^\#}$, $[a_n, b_n] \subset \mathbb{R}_c^\#$ be a hyper infinite sequence of closed intervals satisfying each of the following conditions:

(i) $I_1 \supseteq I_2 \supseteq I_3 \supseteq \dots \supseteq I_n \supseteq \dots$,

(ii) $b_n - a_n \to_{\#} 0$ as $n \to \infty^\#$.

Then $\cap_{n=1}^{\infty} I_n$ consists of exactly one hyperreal number $x \in \mathbb{R}_c^\#$. Moreover both sequences $\{a_n\}$ and $\{b_n\}$ $\#$-converge to $x$.

**Proof.** Note that: (a) the set $A = \{a_n|n \in \mathbb{N}^\#\}$ is bounded or hyperbounded above by $b_1$ and (b) the set $A = \{a_n|n \in \mathbb{N}^\#\}$ is $\le$-admissible above subset of $\mathbb{R}_c^\#$.

By Theorem 8.10 there exists $\sup A$. Let $\xi = \sup A$.

Since $I_n$ are nested, for any positive hyperintegers $m$ and $n$ we have $a_m \le a_{m+n} \le b_{m+n} \le b_n$, so that $\xi \le b_n$ for each $n \in \mathbb{N}^\#$. Since we obviously have $a_n \le \xi$ for each $n \in \mathbb{N}^\#$, we have $a_n \le \xi \le b_n$ for all $n \in \mathbb{N}^\#$, which implies $\xi \in \cap_{n=1}^{\infty} I_n$. Finally, if $\xi, \eta \in \cap_{n=1}^{\infty} I_n$, with $\xi \le \eta$, then we get $0 \le \eta - \xi \le b_n - a_n$, for all $n \in \mathbb{N}^\#$, so that $0 \le \eta - \xi \le \inf_{n \in \mathbb{N}^\#} |b_n - a_n| = 0$.

**Theorem 8.12. (Generalized Squeeze Theorem)**

Let $\{a_n\}$, $\{c_n\}$ be two hyper infinite sequences $\#$-converging to $L$, and $\{b_n\}$ a hyper infinite sequence. If $\forall n \ge K, K \in \mathbb{N}^\#$ we have $a_n \le b_n \le c_n$, then $\{b_n\}$ also $\#$-converges to $L$.

**Proof.** Choose an $\varepsilon > 0, \varepsilon \approx 0$. By definition of the $\#$-limit, there is an $N_1 \in \mathbb{N}^\#$ such that for all $n > N_1$ we have $|a_n - L| < \varepsilon$, in other words $L - \varepsilon < a_n < L + \varepsilon$. Similarly, there is an $N_2 \in \mathbb{N}^\#$ such that for all $n > N_2$ we have $L - \varepsilon < c_n < L + \varepsilon$. Denote $N = \max(N_1, N_2, K)$. Then for $n > N, L - \varepsilon < a_n \le b_n \le c_n < L + \varepsilon$, in other words $|b_n - L| < \varepsilon$. Since $\varepsilon > 0, \varepsilon \approx 0$ was arbitrary, by definition of the $\#$-limit this says that $\#\text{-lim}_{n \to \infty^\#} b_n = L$.

**Theorem 8.13. (Corollary of the Generalized Squeeze Theorem).**

If $\#\text{-lim}_{n \to \infty^\#} |a_n| = 0$ then $\#\text{-lim}_{n \to \infty^\#} a_n = 0$.

**Proof.** We know that $-|a_n| \le a_n \le |a_n|$. We want to apply the Generalized Squeeze Theorem. We are given that $\#\text{-lim}_{n \to \infty^\#} |a_n| = 0$. This also implies that $\#\text{-lim}_{n \to \infty^\#} (-|a_n|) = 0$. So by the Generalized Squeeze Theorem, $\#\text{-lim}_{n \to \infty^\#} a_n = 0$.

**Theorem 8.14. (Generalized Bolzano-Weierstrass Theorem)**
Every hyperbounded hyperinfinite sequence has a #-convergent hyper infinite
subsequence.

**Proof.** Let {$w_n$}$_{n \in \mathbb{N}^\#}$ be a hyperbounded hyper infinite sequence. Then, there exists an interval $[a_1, b_1]$ such that $a_1 \le w_n \le b_1$ for all $n \in \mathbb{N}^\#$.

Either $[a_1, \frac{a_1+b_1}{2}]$ or $[\frac{a_1+b_1}{2}, b_1]$ contains hyper infinitely many terms of $\{w_n\}$. That is, there exists hyperinfinitely many $n$ in $\mathbb{N}^\#$ such that $a_n$ is in $[a_1, \frac{a_1+b_1}{2}]$ or there exists hyper infinitely many $n$ in $\mathbb{N}^\#$ such that $a_n$ is in $[\frac{a_1+b_1}{2}, b_1]$. If $[a_1, \frac{a_1+b_1}{2}]$ contains hyper infinitely many terms of $\{w_n\}$, let $[a_2, b_2] = [a_1, \frac{a_1+b_1}{2}]$. Otherwise, let $[a_2, b_2] = [\frac{a_1+b_1}{2}, b_1]$. Either $[a_2, \frac{a_2+b_2}{2}]$ or $[\frac{a_2+b_2}{2}, b_2]$ contains hyper infinitely many terms of $\{w_n\}_{n \in \mathbb{N}^\#}$. If $[a_2, \frac{a_2+b_2}{2}]$ contains hyper infinitely many terms of $\{w_n\}$, let $[a_3, b_3] = [a_2, \frac{a_2+b_2}{2}]$. Otherwise, let $[a_3, b_3] = [\frac{a_2+b_2}{2}, b_2]$. By hyper infinite induction, we can continue this construction and obtain hyper infinite sequence of intervals $\{[a_n, b_n]\}_{n \in \mathbb{N}^\#}$ such that:

(i) for each $n \in \mathbb{N}^\#$, $[a_n, b_n]$ contains hyper infinitely many terms of $\{w_n\}_{n \in \mathbb{N}^\#}$,

(ii) for each $n \in \mathbb{N}^\#$, $[a_{n+1}, b_{n+1}] \subseteq [a_n, b_n]$ and

(iii) for each $n \in \mathbb{N}^\#$, $b_{n+1} - a_{n+1} = \frac{1}{2}(b_n - a_n)$.

Then generalized nested intervals theorem implies that the intersection of all of the intervals $[a_n, b_n]$ is a single point $w$. We will now construct a hyper infinite subsequence of $\{w_n\}_{n \in \mathbb{N}^\#}$ which will #-converge to $w$.

Since $[a_1, b_1]$ contains hyper infinitely many terms of $\{w_n\}_{n \in \mathbb{N}^\#}$, there exists $k_1 \in \mathbb{N}^\#$ such that $w_{k_1}$ is in $[a_1, b_1]$. Since $[a_2, b_2]$ contains hyper infinitely many terms of $\{w_n\}_{n \in \mathbb{N}^\#}$, there exists $k_2 \in \mathbb{N}^\#$, $k_2 > k_1$, such that $w_{k_2}$ is in $[a_2, b_2]$. Since $[a_3, b_3]$ contains hyper infinitely many terms of $\{w_n\}_{n \in \mathbb{N}^\#}$, there exists $k_3 \in \mathbb{N}^\#$, $k_3 > k_2$, such that $w_{k_3}$ is in $[a_3, b_3]$. Continuing this process by hyper infinite induction, we obtain hyper infinite sequence $\{w_{k_n}\}_{n \in \mathbb{N}^\#}$ such that $w_{k_n} \in [a_n, b_n]$ for each $n \in \mathbb{N}^\#$. The sequence $\{w_{k_n}\}_{n \in \mathbb{N}^\#}$ is a subsequence of $\{w_n\}_{n \in \mathbb{N}^\#}$ since $k_{n+1} > k_n$ for each $n \in \mathbb{N}^\#$. Since $a_n \to w$, and $a_n \le w_n \le b_n$ for each $n \in \mathbb{N}^\#$, the squeeze theorem implies that $w_{k_n} \to w$.

**Definition 8.12.** Let $\{a_n\}$ be a $\mathbb{R}^\#$-valued hyper infinite sequence i.e., $a_n \in \mathbb{R}^\#$, $n \in \mathbb{N}^\#$. Say that $\{a_n\}$ #-tends to 0 if, given any $\varepsilon > 0$, $\varepsilon \approx 0$, there is a hypernatural number $N \in \mathbb{N}^\# \setminus \mathbb{N}$, $N = N(\varepsilon)$ such that, for all $n > N$, $|a_n| \le \varepsilon$. We often denote this symbolically by $a_n \to_\# 0$.

We can also, at this point, define what it means for a hyperreal sequence #-tends to a given number $q \in \mathbb{R}^\#$: $\{a_n\}$ #-tends to $q$ if the hyperreal sequence $\{a_n - q\}$ #-tends to 0 i.e., $a_n - q \to_\# 0$.

**Definition 3.13.** Let $\{a_n\}, n \in \mathbb{N}^\#$ be a hyperreal sequence. We call $\{a_n\}$ a Cauchy hyperreal sequence if the difference between its terms #-tends to 0. To be precise: given any hyperreal number $\varepsilon > 0$, $\varepsilon \approx 0$, there is a hypernatural number $N = N(\varepsilon)$ such that for any $m, n > N, |a_n - a_m| < \varepsilon$.

**Theorem 8.15.** If $\{a_n\}$ is a #-convergent hyperreal sequence (that is, $a_n \to_\# b$ for some hyperreal number $b \in \mathbb{R}^\#$), then $\{a_n\}$ is a Cauchy hyperreal sequence.

**Theorem 8.16.** If $\{a_n\}$ is a Cauchy hyperreal sequence, then it is bounded or hyper bounded; that is, there is some $M \in \mathbb{R}^\#$ such that $|a_n| \le M$ for all $n \in \mathbb{N}^\#$.

**Theorem 8.17.** Any Cauchy hyperreal sequence $\{a_n\}$ has a #-limit in $\mathbb{R}^\#$ i.e., there
exists

$b \in \mathbb{R}_c^\# $ such that $a_n \to b$.

**Proof.** By Definition 8.13 given $\varepsilon > 0, \varepsilon \approx 0$, there is a hypernatural number $N = N(\varepsilon)$ such that for any $n, n' > N$,

$$
|a_n - a_{n'}| < \varepsilon. \tag{8.1}
$$

From (8.1) for any $n, n' > N$ we get

$$
a_{n'} - \varepsilon < a_n < a_n + \varepsilon. \tag{8.2}
$$

The generalized Bolzano-Weierstrass theorem implies there is a #-convergent
hyper infinite subsequence {$a_{n_k}$} $\subset$ {$a_n$} such that $a_{n_k} \to_{\#} b$ for some hyperreal
number $b \in \mathbb{R}_c^\#$. Let us show that the sequence {$a_n$} also #-convergent to this $b \in \mathbb{R}_c^\#$.
We can choose $k \in \mathbb{N}^\#$ so large that $n_k > N$ and

$$
|a_{n_k} - b| < \varepsilon. \tag{8.3}
$$

We choose now in (8.1) $n' = n_k$ and therefore

$$
|a_n - a_{n_k}| < \epsilon. \tag{8.4}
$$

From (8.3) and (8.4) for any $n > N$ we get

$$
|(a_{n_k} - b) + (a_n - a_{n_k})| = |a_n - b| < 2\epsilon. \quad (8.5)
$$

Thus $a_n \to b$ as well.

9.The Extended Hyperreal Number System $\hat{\mathbb{R}}_c^\#$

**Definition 9.1.** (a) A set $S \subset \mathbb{N}^\#$ is hyperfinite if $card(S) = card(\{x|0 \le x \le n\})$,
where $n \in \mathbb{N}^\# \setminus \mathbb{N}$. (b) A set $S \subseteq \mathbb{N}^\#$ is hyper infinite if $card(S) = card(\mathbb{N}^\#)$

**Notation 9.1.** If $F$ is an arbitrary collection of sets, then $\cup\{S|S \in F\}$ is the set of all elements that are members of at least one of the sets in $F$, and $\cap\{S|S \in F\}$ is the set of all elements that are members of every set in $F$. The union and intersection of finitely or hyper finitely many sets $S_k, 0 \le k \le n \in \mathbb{N}^\#$ are also written as $\cup_{k=0}^n S_k$ and $\cap_{k=0}^n S_k$. The union and intersection of an hyperinfinite sequence $S_k, k \in \mathbb{N}^\#$ of sets are written as $\cup_{k=0}^\infty S$ or $\cup_{n \in \mathbb{N}^\#} S$ and $\cap_{k=0}^\infty S$ or $\cap_{n \in \mathbb{N}^\#} S$ correspondingly.

A nonempty set $S$ of hyperreal numbers $\mathbb{R}_c^\#$ is unbounded above if it has no hyperfinite
upper bound, or unbounded below if it has no hyperfinite lower bound. It is convenient
to adjoin to the hyperreal number system two points, $+\infty^\#$ (which we also write more
simply as $\infty^\#$) and $-\infty^\#$, and to define the order relationships between them and any
hyperreal number $x \in \mathbb{R}_c^\#$ by $-\infty^\# < x < \infty^\#$.

We call $-\infty^\#$ and $\infty^\#$ points at hyperinfinity. If $S$ is a nonempty set of hyperreals, we
write $\sup S = \infty^\#$$^\#$ to indicate that $S$ is hyper unbounded above, and $\inf S = -\infty^\#$$^\#$ to
indicate that $S$ is hyper unbounded below.

\#-Open and \#-Closed Sets on $\hat{\mathbb{R}}_c^\#$.

**Definition 9.2.** If $a$ and $b$ are in the extended hyperreals and $a < b$, then the \#-open interval $(a, b)$ is defined by $(a, b) \triangleq \{x|a < x < b\}$.

The \#-open intervals $(a, \infty^\#$) and $(-\infty^\#, b)$ are semi-hyper infinite if $a$ and $b$ are finite or hyperfinite, and $(-\infty^\#, \infty^\#)$ is the entire hyperreal line.

If $-\infty^\# < a < b < \infty^\#$, the set $[a, b] \triangleq \{x|a \le x \le b\}$ is \#-closed, since its complement is the union of the \#-open sets $(-\infty^\#, a)$ and $(b, \infty^\#)$. We say that $[a, b]$ is a \#-closed
interval. Semi-hyper infinite #-closed intervals are sets of the form $[a, \infty) = \{x|a \le x\}$ and $(-\infty^#, a] = \{x|x \le a\}$, where $a$ is finite or hyperfinite. They are #-closed sets, since their complements are the #-open intervals $(-\infty^#, a)$ and $(a, \infty^#)$, respectively.

**Definition 9.3.** If $x_0 \in \mathbb{R}_c^#$ is a hyperreal number and $\varepsilon > 0, \varepsilon \approx 0$ then the #-open interval $(x_0 - \varepsilon, x_0 + \varepsilon)$ is an #-neighborhood of $x_0$. If a set $S \subset \mathbb{R}_c^#$ contains an #-neighborhood of $x_0$, then $S$ is a #-neighborhood of $x_0$, and $x_0$ is an #-interior point of $S$. The set of #-interior points of $S$ is the #-interior of $S$, denoted by $\text{-Int}(S)$.

(i) If every point of $S$ is an #-interior point (that is, $S = \text{-Int}(S)$), then $S$ is #-open.

(ii) A set $S$ is #-closed if $S^c = \mathbb{R}_c^# \setminus S$ is #-open.

**Example 9.1.** An open interval $(a, b)$ is an #-open set, because if $x_0 \in (a, b)$ and $\varepsilon \le \min \{x_0 - a; b - x_0\}$, then $(x_0 - \varepsilon, x_0 + \varepsilon) \subset (a, b)$.

**Remark 9.1.** The entire hyperline $\hat{\mathbb{R}}_c^# = (-\infty^#, \infty^#)$ is #-open, and therefore $\emptyset$ is #-closed.

However, $\emptyset$ is also #-open, for to deny this is to say that $\emptyset$ contains a point that is not an #-interior point, which is absurd because $\emptyset$ contains no points. Since $\emptyset$ is #-open, $\hat{\mathbb{R}}_c^#$ is #-closed. Thus, $\hat{\mathbb{R}}_c^#$ and $\emptyset$ are both #-open and #-closed.

**Remark 9.2.** They are not the only subsets of $\hat{\mathbb{R}}_c^#$ with this property mentioned above.

**Definition 9.4.** A deleted #-neighborhood of a point $x_0$ is a set that contains every point of some #-neighborhood of $x_0$ except for $x_0$ itself. For example, $S = \{x|0 < |x - x_0| < \varepsilon\}$, where $\varepsilon \approx 0$, is a deleted #-neighborhood of $x_0$. We also say that it is a deleted $\varepsilon$-#-neighborhood of $x_0$.

**Theorem 9.1.** (a) The union of #-open sets is #-open:

(b) The #-intersection of #-closed sets is #-closed:

These statements apply to arbitrary collections, hyperfinite or hyperinfinite, of #-open and #-closed sets.

**Proof** (a) Let $L$ be a collection of #-open sets and $S = \bigcup \{G | G \in L\}$.

If $x_0 \in S$, then $x_0 \in G_0$ for some $G_0$ in $L$, and since $G_0$ is #-open, it contains some $\varepsilon$-#-neighborhood of $x_0$. Since $G_0 \subset S$, this $\varepsilon$-#-neighborhood is in $S$, which is consequently a #-neighborhood of $x_0$. Thus, $S$ is a #-neighborhood of each of its points, and therefore #-open, by definition.

(b) Let $F$ be a collection of #-closed sets and $T = \bigcap \{H | H \in F\}$. Then $T^c = \bigcup \{H^c | H \in F\}$ and, since each $H^c$ is #-open, $T^c$ is #-open, from (a). Therefore, $T$ is #-closed, by definition.

**Example 9.2.** If $-\infty^# < a < b < \infty^#$, the set $[a, b] = \{x|a \le x \le b\}$ is #-closed, since its complement is the union of the #-open sets $(-\infty^#a)$ and $(b, \infty^#)$. We say that $[a, b]$ is a #-closed interval. The set $[a, b) = \{x|a \le x < b\}$ is a half-#-closed or half-#-open interval if $-\infty^# < a < b < \infty^#$, as is $(a, b] = \{x|a < x \le b\}$ however, neither of these sets is #-open or #-closed. Semi-infinite #-closed intervals are sets of the form $[a, \infty^#) = \{x|a \le x\}$ and $(-\infty^#, a] = \{x|x \le a\}$, where $a$ is hyperfinite. They are #-closed sets, since their complements are the #-open intervals $(-\infty^#, a)$ and

$(a, \infty^#)$, respectively.

**Definition 9.5.** Let $S$ be a subset of $\hat{\mathbb{R}}_c^# = (-\infty^#, \infty^#)$. Then

(a) $x_0$ is a #-limit point of $S$ if every deleted #-neighborhood of $x_0$ contains a point of $S$.

(b) $x_0$ is a boundary point of $S$ if every #-neighborhood of $x_0$ contains at least one point in $S$ and one not in $S$. The set of #-boundary points of $S$ is the #-boundary of $S$,
denoted

by #-∂S. The #-closure of S, denoted by #-S, is S ∪ #-∂S.

(c) $x_0$ is an #-isolated point of S if $x_0 \in S$ and there is a #-neighborhood of $x_0$ that contains

no other point of S.

(d) $x_0$ is #-exterior to S if $x_0$ is in the #-interior of $S^c$. The collection of such points is the
#-exterior of S.

**Theorem 9.2.** A set S is #-closed if and only if no point of $S^c$ is a #-limit point of S.

**Proof.** Suppose that S is #-closed and $x_0 \in S^c$. Since $S^c$ is #-open, there is a
#-neighborhood of $x_0$ that is contained in $S^c$ and therefore contains no points of S.

Hence, $x_0$ cannot be a #-limit point of S. For the converse, if no point of $S^c$
is a #-limit point of S then every point in $S^c$ must have a #-neighborhood contained
in $S^c$. Therefore, $S^c$ is #-open and S is #-closed.

**Corollary 9.1.** A set S is #-closed if and only if it contains all its #-limit points.

If S is #-closed and hyper bounded, then inf(S) and sup(S) are both in S.

**Proposition 9.1.** If S is #-closed and hyper bounded, then inf(S) and sup(S) are both in S.

#-Open Coverings

**Definition 9.6.** A collection $H$ of #-open sets of $\mathbb{R}^\#_c$ is an #-open covering of a set $S$ if every point in $S$ is contained in a set $H$ belonging to $H$; that is, if $S \subset \cup \{F | F \in H\}$.

**Definition 9.7.** A set $S \subset \mathbb{R}^\#_c$ is called #-compact (or hyper compact) if each of its
#-open covers has a finite or hyperfinite subcover.

**Theorem 9.3. (Generalized Heine–Borel Theorem)** If $H$ is an #-open covering of a
#-closed and hyper bounded subset $S$ of the hyperreal line $\mathbb{R}^\#_c$ (or of the $\mathbb{R}^{\#n}_c$, $n \in \mathbb{N}^\#$)
then $S$ has an #-open

covering $\tilde{H}$ consisting of hyper finite many #-open sets belonging to $H$.

**Proof.** If a set $S$ in $\mathbb{R}^{\#n}_c$ is hyper bounded, then it can be enclosed within an $n$-box
$T_0 = [-a, a]^n$ where $a > 0$. By the property above, it is enough to show that $T_0$ is
#-compact.

Assume, by way of contradiction, that $T_0$ is not #-compact. Then there exists an hyper
infinite open cover $C_{\infty#}$ of $T_0$ that does not admit any hyperfinite subcover. Through
bisection of each of the sides of $T_0$, the box $T_0$ can be broken up into $2n$ sub $n$-boxes,
each of which has diameter equal to half the diameter of $T_0$. Then at least one of the
$2n$ sections of $T_0$ must require an hyper infinite subcover of $C_{\infty#}$, otherwise $C_{\infty#}$ itself
would have a hyperfinite subcover, by uniting together the hyperfinite covers of the
sections. Call this section $T_1$. Likewise, the sides of $T_1$ can be bisected, yielding $2n$
sections of $T_1$, at least one of which must require an hyper infinite subcover of $C_{\infty#}$.

Continuing in like manner yields a decreasing hyper infinite sequence of nested $n$-boxes:
$T_0 \supset T_1 \supset T_2 \supset \dots \supset T_k \supset \dots$, $k \in \mathbb{N}^\#$, where the side length of $T_k$ is $(2a)/2^k$, which
#-converges to 0 as $k$ tends to hyper infinity, $k \to \infty^\#$. Let us define a hyper infinite
sequence $\{x_k\}_{k \in \mathbb{N}^\#}$ such that each $x_k : x_k \in T_k$. This hyper infinite sequence is Cauchy,
so it must #-converge to some #-limit $L$. Since each $T_k$ is #-closed, and for each $k$ the
sequence $\{x_k\}_{k \in \mathbb{N}^\#}$ is eventually always inside $T_k$, we see that $L \in T_k$ for each $k \in \mathbb{N}^\#$.

Since $C_{\infty#}$ covers $T_0$, then it has some member $U \in C_{\infty#}$ such that $L \in U$. Since U is
open, there is an $n$-ball $B(L) \subseteq U$. For large enough $k$, one has $T_k \subseteq B(L) \subseteq U$, but then
the infinite number of members of $C_\infty$ needed to cover $T_k$ can be replaced by just one: $U$, a contradiction. Thus, $T_0$ is #-compact. Since $S$ is #-closed and a subset of the #-compact set $T_0$, then $S$ is also #-compact.

As an application of the Generalized Heine–Borel theorem, we give a short proof of the Generalized Bolzano–Weierstrass Theorem.

**Theorem 9.4.** (Generalized Bolzano–Weierstrass Theorem) Every hyper bounded hyper infinite set $S \subset \mathbb{R}_c^\#$ has at least one #-limit point.

**Proof.** We will show that a hyper bounded nonempty set without a #-limit point can contain only finite or a hyper finite number of points. If $S$ has no #-limit points, then $S$ is #-closed (Theorem 9.) and every point $x \in S$ has an w-#-open neighborhood $N_x$ that contains no point of $S$ other than $x$. The collection $H = \{N_x|x \in S\}$ is an w-#-open covering for $S$. Since $S$ is also hyper bounded, Theorem 9.3 implies that $S$ can be covered by finite or a hyper finite collection of sets from $H$, say $N_{x_1},...,N_{x_n},n \in \mathbb{N}^\#$. Since these sets contain only $x_1,...,x_n$ from $S$, it follows that $S = \{x_k\}_{1\le k \le n},n \in \mathbb{N}^\#$.

# 10. External non-Archimedean field $\mathbb{R}_c^\#$.

## 10.1. External non-Archimedean field $\mathbb{R}_c^\#$ via Cauchy completion of internal non-Archimedean field $\mathbb{R}$.

**Definition 10.1.** A hyper infinite sequence of hyperreal numbers from $\mathbb{R}$ is a function $a : \mathbb{N}^\# \to \mathbb{R}$ from hypernatural numbers $\mathbb{N}^\#$ into the hyperreal numbers $\mathbb{R}$.

We usually denote such a function by $n \mapsto a_n$, or by $a : n \to a_n$, so the terms in the sequence are written $\{a_1, a_2, a_3, ..., a_n\}$. To refer to the whole hyper infinite sequence, we will write $\{a_n\}_{n=1}^{\infty}$, or $\{a_n\}_{n \in \mathbb{N}^\#}$, or for the sake of brevity simply $\{a_n\}$.

**Definition 10.2.** Let $\{a_n\}$ be a hyper infinite $\mathbb{R}$-valued sequence mentioned above. Say that $\{a_n\}$ #-tends to 0 if, given any $\varepsilon > 0$, $\varepsilon \approx 0$, there is a hypernatural number $N \in \mathbb{N}^\#\mathbb{N}$, $N = N(\varepsilon)$ such that, after $N$ (i.e. for all $n > N$), $|a_n| \le \varepsilon$. We denote this symbolically by $a_n \to_\# 0$.

We can also, at this point, define what it means for a hyper infinite $\mathbb{R}$-valued sequence #-tends to any given number $q \in \mathbb{R} : \{a_n\}$ #-tends to $q$ if the hyper infinite sequence $\{a_n - q\}$ #-tends to 0 i.e., $a_n - q \to_\# 0$.

**Definition 10.3.** Let $\{a_n\}$ be a hyper infinite $\mathbb{R}$-valued sequence. We call $\{a_n\}$ a Cauchy hyper infinite $\mathbb{R}$-valued sequence if the difference between its terms #-tends to 0. To be precise: given any hyperreal number such that $\varepsilon > 0$, $\varepsilon \approx 0$, there is a hypernatural number $N = N(\varepsilon)$ such that for any $m, n > N$, $|a_n - a_m| < \varepsilon$.

**Theorem 10.1.** If $\{a_n\}$ is a #-convergent hyper infinite $\mathbb{R}$-valued sequence (that is, $a_n \to_\# q$ for some hyperreal number $q \in \mathbb{R}$), then $\{a_n\}$ is a Cauchy hyper infinite $\mathbb{R}$-valued sequence.

**Proof.** We know that $a_n \to_\# q$. Here is a ubiquitous trick: instead of using $\varepsilon$ in the definition Definition 10.3, start with an arbitrary infinite small $\varepsilon > 0$, $\varepsilon \approx 0$ and then choose $N \in \mathbb{N}^\#\mathbb{N}$ so that $|a_n - q| < \varepsilon/2$ when $n > N$. Then if $m, n > N$, we have $|a_n - a_m| = |(a_n - q) - (a_m - q)| \le |a_n - q| + |a_m - q| < \varepsilon/2 + \varepsilon/2 = \varepsilon$. This shows that $\{a_n\}_{n \in \mathbb{N}^\#}$ is a Cauchy sequence.

**Theorem 10.2.** If $\{a_n\}$ is a Cauchy hyper infinite $\mathbb{R}$-valued sequence, then it is bounded or hyper bounded; that is, there is some finite or hyperfinite $M \in \mathbb{R}$ such
that $|a_n|\le M$ for all $n \in \mathbb{N}^\#$.

**Proof.** Since $\{a_n\}$ is Cauchy, setting $\varepsilon = 1$ we know that there is some $N \in \mathbb{N}^\#$ such that $|a_m - a_n| < 1$ whenever $m, n > N$. Thus, $|a_{N+1} - a_n| < 1$ for $n > N$. We can rewrite this as $a_{N+1} - 1 < a_n < a_{N+1} + 1$. This means that $|a_n|$ is less than the maximum of $|a_{N+1} - 1|$ and $|a_{N+1} + 1|$. So, set $M$ equal to the maximum number in the following list: $\{|a_0|, |a_1|, \dots, |a_N|, |a_{N+1} - 1|, |a_{N+1} + 1|\}$. Then for any term $a_n$, if $n \le N$, then $|a_n|$ appears in the list and so $|a_n|\le M$; if $n > N$, then (as shown above) $|a_n|$ is less than at least one of the last two entries in the list, and so $|a_n|\le M$. Hence, $M \in *\mathbb{R}$ is a bound for the sequence $\{a_n\}$.

**Definition 10.4.** Let $S$ be a set. A relation $x \sim y$ among pairs of elements of $S$ is said to be an equivalence relation if the following three properties hold:
* Reflexivity: for any $s \in S, s \sim s$.
* Symmetry: for any $s, t \in S$, if $s \sim t$ then $t \sim s$.
* Transitivity: for any $s, t, r \in S$, if $s \sim t$ and $t \sim r$, then $s \sim r$.

**Theorem 10.3.** Let $S$ be a set, with an equivalence relation $(\cdot \sim \cdot)$ on pairs of elements. For $s \in S$, denote by $cl[s]$ the set of all elements in $S$ that are related to $s$. Then for any $s, t \in S$, either $cl[s] = cl[t]$ or $cl[s]$ and $cl[t]$ are disjoint. The hyperreal numbers $*\mathbb{R}^\#_c$ will be constructed as equivalence classes of Cauchy hyper infinite $*\mathbb{R}$-valued sequences. Let $F_{*\mathbb{R}}$ denote the set of all Cauchy hyper infinite $*\mathbb{R}$-valued sequences of hyperreal numbers. We define the equivalence relation on $F_{*\mathbb{R}}$.

**Definition 10.5.** Let $\{a_n\}$ and $\{b_n\}$ be in $F_{*\mathbb{R}}$. Say they are #-equivalent if $a_n - b_n \to_\# 0$ i.e., if and only if the hyper infinite $*\mathbb{R}$-valued sequence $\{a_n - b_n\}$ tends to 0.

**Theorem 10.4.** Definition 10.5 yields an equivalence relation on $F_{*\mathbb{R}}$.

**Proof.** We need to show that this relation is reflexive, symmetric, and transitive.
*Reflexive:* $a_n - a_n = 0$, and the hyper infinite sequence all of whose terms are 0 clearly #-converges to 0. So $\{a_n\}$ is related to $\{a_n\}$.

*Symmetric:* Suppose $\{a_n\}$ is related to $\{b_n\}$, so $a_n - b_n \to_\# 0$.

But $b_n - a_n = -(a_n - b_n)$, and since only the absolute value $|a_n - b_n| = |b_n - a_n|$ comes into play in Definition 10.2, it follows that $b_n - a_n \to_\# 0$ as well. Hence, $\{b_n\}$ is related to $\{a_n\}$.

*Transitive:* Here we will use the $\varepsilon/2$ trick we applied to prove Theorem 10.1. Suppose $\{a_n\}$ is related to $\{b_n\}$, and $\{b_n\}$ is related to $\{c_n\}$. This means that $a_n - b_n \to_\# 0$ and $b_n - c_n \to_\# 0$. To be fully precise, let us fix $\varepsilon > 0$, $\varepsilon \approx 0$; then there exists an $N \in \mathbb{N}^\#$ such that for all $n > N$, $|a_n - b_n| < \varepsilon/2$; also, there exists an $M$ such that for all $n > M$, $|b_n - c_n| < \varepsilon/2$. Well, then, as long as $n$ is bigger than both $N$ and $M$, we have that $|a_n - c_n| \le |(a_n - b_n) + (b_n - c_n)| \le |a_n - b_n| + |b_n - c_n| < \varepsilon/2 + \varepsilon/2 = \varepsilon$.

So, choosing $L$ equal to the max of $N, M$, we see that given $\varepsilon > 0$ we can always choose $L$ so that for $n > L$, $|a_n - c_n| < \varepsilon$. This means that $a_n - c_n \to_\# 0$ – i.e. $\{a_n\}$ is related to $\{c_n\}$.

**Definition 10.6.** The external hyperreal numbers $*\mathbb{R}^\#_c$ are the equivalence classes $cl[\{a_n\}]$ of Cauchy hyper infinite $*\mathbb{R}$-valued sequences of hyperreal numbers, as per Definition 10.5. That is, each such equivalence class is an external hyperreal number.

**Definition 10.7.** Given any hyperreal number $q \in *\mathbb{R}$, define a hyperreal number $q^\#$ to be the equivalence class of the hyper infinite $*\mathbb{R}$-valued sequence $q^\# = (q, q, q, q, \dots)$
some Cauchy sequence of hyperreal numbers such that for some $N \in \mathbb{N}^{\#}$, $a_n > 0$ for all $n > N$. Given two hyperreal numbers $s, t$, say that $s > t$ if $s - t$ is positive.

**Theorem 10.7.** Let $s, t \in *\mathbb{R}_c^\#$ be hyperreal numbers such that $s > t$, and let $r \in *\mathbb{R}_c^\#$. Then $s + r > t + r$.

**Proof.** Let $s = \text{cl}[\{a_n\}]$, $t = \text{cl}[\{b_n\}]$, and $r = \text{cl}[\{c_n\}]$. Since $s > t$ i.e., $s - t > 0$, we know that there is an $N \in \mathbb{N}^\#$ such that, for $n > N$, $a_n - b_n > 0$. So $a_n > b_n$ for $n > N$. Now, adding $c_n$ to both sides of this inequality (as we know we can do for hyperreal numbers $*\mathbb{R}$), we have $a_n + c_n > b_n + c_n$ for $n > N$, or $(a_n + c_n) - (b_n + c_n) > 0$ for $n > N$. Note also that $(a_n + c_n) - (b_n + c_n) = a_n - b_n$ does not #-converge to 0, by the assumption that $s - t > 0$. Thus, by Definition 10.8, this means that $s + r = \text{cl}[\{a_n + c_n\}] > \text{cl}[\{b_n + c_n\}] = t + r$.

**Theorem 10.8.** Let $s, t \in *\mathbb{R}_c^\#$, $s > 0$ be hyperreal numbers. Then there is $m \in \mathbb{N}^\#$ such that $m \times s > t$.

**Proof.** Let $s, t > 0$ be hyperreal numbers. We need to find a natural number $m$ so that $m \times s > t$. First, recall that, by $m$ in this context, we mean $\text{cl}[\{m, m, m, \dots\}]$. So, letting $s = \text{cl}[\{a_n\}]$ and $t = \text{cl}[\{b_n\}]$, what we need to show is that there exists $m$ with $\text{cl}[\{m, m, m, \dots\}] \times \text{cl}[\{a_1, a_2, a_3, a_4, \dots\}] = \text{cl}[\{m \times a_1, m \times a_2, m \times a_3, m \times a_4, \dots\}] > \text{cl}[\{b_1, b_2, b_3, b_4, \dots\}]$.

Now, to say that $\text{cl}[\{m \times a_n\}] > \text{cl}[\{b_n\}]$, or $\text{cl}[\{m \times a_n - b_n\}]$ is positive, is, by Definition 10.9, just to say that there is $N \in \mathbb{N}^\#$ such that $m \times a_n - b_n > 0$ for all $n > N$, while $m \times a_n - b_n \to_\# 0$. To be precise, the first statement is:
There exist $m, N \in \mathbb{N}^\#$ so that $m \times a_n > b_n$ for all $n > N$.

To produce a contradiction, we assume this is not the case; assume that (#) for every $m$ and $N$, there exists an $n > N$ so that $m \times a_n \le b_n$.

Now, since $\{b_n\}$ is a Cauchy sequence, by Theorem 10.2 it is hyperbounded - there is a hyperreal number $M \in *\mathbb{R}$ such that $b_n \le M$ for all $n \in \mathbb{N}^\#$. Now, by the properties for the hyperreal numbers $*\mathbb{R}$, given any hyperreal number such that $\varepsilon > 0$, $\varepsilon \approx 0$, there is an $m \in \mathbb{N}^\#$ such that $M/m < \varepsilon/2$. Fix such an $m$. Then if $m \times a_n \le b_n$, we have $a_n \le b_n/m \le M/m < \varepsilon/2$.

Now, $\{a_n\}$ is a Cauchy sequence, and so there exists $N$ so that for $n, k > N$, $|a_n - a_k| < \varepsilon/2$.

By Asumption (#), we also have an $n > N$ such that $m \times a_n \le b_n$, which means that $a_n < \varepsilon/2$. But then for every $k > N$, we have that $a_k - a_n < \varepsilon/2$, so $a_k < a_n + \varepsilon/2 < \varepsilon/2 + \varepsilon/2 = \varepsilon$. Hence, $a_k < \varepsilon$ for all $k > N$. This proves that $a_k \to_\# 0$, which by Definition 10.9 contradicts the fact that $\text{cl}[\{a_n\}] = s > 0$.

Thus, there is indeed some $m \in N$ so that $m \times a_n - b_n > 0$ for all sufficiently infinite large $n \in \mathbb{N}^\#\mathbb{N}$. To conclude the proof, we must also show that $m \times a_n - b_n \to 0$. Actually, it is possible that $m \times a_n - b_n \to 0$ (for example if $\{a_n\} = \{1, 1, 1, \dots\}$ and $\{b_n\} = \{m, m, m, \dots\}$). But that's okay: then we can simply choose a larger $m$. That is: let $m$ be a hypernatural number constructed as above, so that $m \times a_n - b_n > 0$ for all sufficiently large $n \in \mathbb{N}^\#\mathbb{N}$. If it happens to be true that $m \times a_n - b_n \to 0$, then the proof is complete.

If, on the other hand, it turned out that $m \times a_n - b_n \to 0$, then take instead the integer $m + 1$. Since $s = \text{cl}[\{a_n\}] > 0$, we have an $n > 0$ for all infinite large $n$, so $(m + 1) \times a_n - b_n = m \times a_n - b_n + a_n > a_n > 0$ for all infinite large $n$, so $m + 1$ works just as well as $m$ did in this regard; and since $m \times a_n - b_n \to 0$, we have
$(m+1) \times a_n - b_n = (m \times a_n - b_n) + a_n \Rightarrow 0$ since $s = \text{cl}[(a_n)] > 0$ (so $a_n \Rightarrow 0$).

It will be handy to have one more Theorem about how the hyperreals $*\mathbb{R}$ and hyperreals $*\mathbb{R}_c^#$ compare before we proceed. This theorem is known as the density of $*\mathbb{R}$ in $*\mathbb{R}_c^#$, and it follows almost immediately from the construction of the $*\mathbb{R}_c^#$ from $*\mathbb{R}$.

**Theorem 10.9.** Given any hyperreal number $r \in *\mathbb{R}_c^#$, and any hyperreal number $\varepsilon > 0$, $\varepsilon \approx 0$, there is a hyperreal number $q \in *\mathbb{R}$ such that $|r-q| < \varepsilon$.

**Proof.** The hyperreal number $r$ is represented by a Cauchy $*\mathbb{R}$-valued sequence $\{a_n\}$. Since this sequence is Cauchy, given $\varepsilon > 0, \varepsilon \approx 0$, there is $N \in \mathbb{N}^#$ so that for all $m, n > N$,

$|a_n - a_m| < \varepsilon$. Picking some fixed $l > N$, we can take the hyperreal number $q$ given by $q = \text{cl}[(a_1, a_2, a_3, \dots)]$. Then we have $r-q = \text{cl}[(a_n - a_l)_{n \in \mathbb{N}^#}]$, and $q-r = \text{cl}[(a_l - a_n)_{n \in \mathbb{N}^#}]$.

Now, since $l > N$, we see that for $n > N, a_n - a_l < \varepsilon$ and $a_l - a_n < \varepsilon$, which means by Definition 10.9 that $r-q < \varepsilon$ and $q-r < \varepsilon$; hence, $|r-q| < \varepsilon$.

**Definition 10.10.** Let $S \subseteq *\mathbb{R}_c^#$ be a non-empty set of hyperreal numbers.

A hyperreal number $x \in *\mathbb{R}_c^#$ is called an upper bound for $S$ if $x \ge s$ for all $s \in S$.

A hyperreal number $x$ is the least upper bound (or supremum $\sup S$) for $S$ if $x$ is an upper bound for $S$ and $x \le y$ for every upper bound $y$ of $S$.

**Remark 10.1.** The order $\le$ given by Definition 10.9 obviously is $\le$-incomplete.

**Definition 10.11.** Let $S \subseteq *\mathbb{R}_c^#$ be a nonempty subset of $*\mathbb{R}_c^#$. We will say that:

(1) $S$ is $\le$-admissible above if the following conditions are satisfied:

(i) $S$ bounded or hyperbounded above;

(ii) let $A(S)$ be a set $\forall x[x \in A(S) \Leftrightarrow x \ge S]$ then for any $\varepsilon > 0, \varepsilon \approx 0$ there exist $\alpha \in S$ and $\beta \in A(S)$ such that $\beta - \alpha \le \varepsilon \approx 0$.

(2) $S$ is $\le$-admissible below if the following condition are satisfied:

(i) $S$ bounded below;

(ii) let $L(S)$ be a set $\forall x[x \in L(S) \Leftrightarrow x \le S]$ then for any $\varepsilon > 0, \varepsilon \approx 0$ there exist $\alpha \in S$ and $\beta \in L(S)$ such that $\alpha - \beta \le \varepsilon \approx 0$.

**Theorem 10.10.** (i) Any $\le$-admissible above subset $S \subset *\mathbb{R}_c^#$ has the least upper bound property. (ii) Any $\le$-admissible below subset $S \subset *\mathbb{R}_c^#$ has the greatest lower bound property.

**Proof.** Let $S \subset *\mathbb{R}_c^#$ be a nonempty subset, and let $M$ be an upper bound for $S$. We are going to construct two sequences of hyperreal numbers, $\{u_n\}$ and $\{l_n\}$. First, since $S$ is nonempty, there is some element $s_0 \in S$. Now, we go through the following hyperinductive procedure to produce numbers $u_0, u_1, u_2, \dots, u_n, \dots$ and $l_1, l_2, l_3, \dots, l_n, \dots$

(i) Set $u_0 = M$ and $l_0 = s$.

(ii) Suppose that we have already defined $u_n$ and $l_n$. Consider the number $m_n = (u_n + l_n)/2$, the average between $u_n$ and $l_n$.

(1) If $m_n$ is an upper bound for $S$, define $u_{n+1} = m_n$ and $l_{n+1} = l_n$.

(2) If $m_n$ is not an upper bound for $S$, define $u_{n+1} = u_n$ and $l_{n+1} = l_n$.

**Remark 10.1.** Since $s < M$, it is easy to prove by hyper infinite induction that

(i) $\{u_n\}$ is a non-increasing sequence: $u_{n+1} \le u_n, n \in \mathbb{N}^#$ and $\{l_n\}$ is a non-decreasing sequence $l_{n+1} \ge l_n, n \in \mathbb{N}^#$, (ii) $u_n$ is an upper bound for $S$ for all $n \in \mathbb{N}^#$ and $l_n$ is never an upper bound for $S$ for any $n \in \mathbb{N}^#$.
This gives us the following lemma.

**Lemma 10.2.** {$u_n$} and {$l_n$} are Cauchy *$\mathbb{R}$*-valued sequences of hyperreal numbers.

**Proof.** Note that each $l_n \le M$ for all $n \in \mathbb{N}^#$. Since {$l_n$} is non-decreasing, it follows that {$l_n$} is Cauchy. For {$u_n$}, we have $u_n \ge s_0$ for all $n \in \mathbb{N}^#$, and so $-u_n \le -s_0$. Since {$u_n$} is non-increasing, {$-u_n$} is non-decreasing, and so as above, {$-u_n$} is Cauchy. It is easy to verify that, therefore, {$u_n$} is Cauchy.

The following Lemma shows that {$u_n$} does #-tend to a hyperreal number $u \in *\mathbb{R}_c^#$.

**Lemma 10.3.** There is a hyperreal number $u \in *\mathbb{R}_c^#$ such that $u_n \to_{\#} u$.

**Proof.** Fix a term $u_n$ in the sequence {$u_n$}. By Theorem 10.9, there is a hyperreal number $q_n \in *\mathbb{R}$, $n \in \mathbb{N}^#$ such that $|u_n - q_n| < 1/n$. Consider the sequence {$q_1, q_2, q_3, \dots, q_n, \dots$} of hyperreal numbers. We will show this sequence is Cauchy. Fix $\varepsilon > 0$, $\varepsilon \approx 0$. By the Theorem 10.8, we can choose $N \in \mathbb{N}^#$ so that $1/N < \varepsilon/3$. We know, since {$u_n$} is Cauchy, that there is an $M \in \mathbb{N}^#$ such that for $n, m > M$, $|u_n - u_m| < \varepsilon/3$. Then, so long as $n, m > \max\{N, M\}$, we have

$$|q_n - q_m| \le |(q_n - u_n) + (u_n - u_m) + (u_m - q_m)| \le |q_n - u_n| + |u_n - u_m| + |u_m - q_m| < \varepsilon/3 + \varepsilon/3 + \varepsilon/3 = \varepsilon.$$

Thus, {$q_n$} is a Cauchy sequence of internal hyperreal numbers, and so it represents the external hyperreal number $u = \text{cl}[\{q_n\}]$. We must show that $u_n - u \to_{\#} 0$, but this is practically built into the definition of $u$. To be precise, letting $q_n^*$ be the hyperreal number

$\text{cl}[\{q_n, q_n, q_n, \dots\}]$, we see immediately that $q_n^* - u \to_{\#} 0$ (this is precisely equivalent to the statement that {$q_n$} is Cauchy). But $u_n - q_n^* < 1/n$ by construction; it is easily verify that the assertion that if a sequence $q_n^* \to_{\#} u$ and $u_n - q_n^* \to_{\#} 0$, then $u_n \to_{\#} u$. So {$u_n$}, a non-increasing sequence of upper bounds for $S$, tends to a hyperreal number $u$. As you've guessed, $u$ is the least upper bound of our set $S$. To prove this, we need one more lemma.

**Lemma 10.4.** $l_n \to_{\#} u$.

**Proof.** First, note in the first case above, we have that $u_{n+1} - l_{n+1} = m_n - l_n = \frac{u_n + l_n}{2} - l_n = \frac{u_n - l_n}{2}$. In the second case, we also have

$$u_{n+1} - l_{n+1} = u_n - m_n = u_n - \frac{u_n + l_n}{2} = \frac{u_n - l_n}{2}.$$

$$u_{n+1} - l_{n+1} = u_n - m_n = u_n - \frac{u_n + l_n}{2} = \frac{u_n - l_n}{2}.$$

Now, this means that $u_1 - l_1 = \frac{1}{2}(M - s)$, and so $u_2 - l_2 = \frac{1}{2}(u_1 - l_1) = \frac{1}{2^2}(L - s)$, and in general by hyperinfinite induction, $u_n - l_n = 2^{-n}(L - s)$. Since $L > s$ so $L - s > 0$, and since $2^{-n} < 1/n$, by the Theorem 10.8, we have for any $\varepsilon > 0$ that $2^{-n}(L - s) < \varepsilon$ for all sufficiently large $n \in \mathbb{N}^#$. Thus, $u_n - l_n < 2^{-n}(L - s) < \varepsilon$ as well, and so $u_n - l_n \to_{\#} 0$. Again, it is easily verify that, since $u_n \to_{\#} u$, we have $l_n \to_{\#} u$ as well.

**Remark 10.2.** Note that assumption in Theorem 10.10 that $S$ is $\le$-admissible above subset of $\mathbb{R}_c^#$ is necessarily, othervice Theorem 10.10 is not holds.

**Theorem 10.11. (Generalized Nested Intervals Theorem)**

Let {$I_n$}$_{n \in \mathbb{N}^#}$ = {[${a_n, b_n}$]}$_{n \in \mathbb{N}^#}$, [$a_n, b_n$] $\subset \mathbb{R}_c^#$ be a hyper infinite sequence of closed intervals satisfying each of the following conditions:
(i) $I_1 \supseteq I_2 \supseteq I_3 \supseteq \dots \supseteq I_n \supseteq \dots,$

(ii) $b_n - a_n \to_\# 0$ as $n \to \infty$.

Then $\cap_{n=1}^\infty I_n$ consists of exactly one hyperreal number $x \in \mathbb{R}_c^\#$. Moreover both sequences $\{a_n\}$ and $\{b_n\}$ #-converge to $x$.

**Proof.** Note that: (a) the set $A = \{a_n|n \in \mathbb{N}^\#\}$ is hyperbounded above by $b_1$ and (b) the set $A = \{a_n|n \in \mathbb{N}^\#\}$ is $\le$-admissible above subset of $\mathbb{R}_c^\#$.

By Theorem 10.10 there exists $\sup A$. Let $\xi = \sup A$.

Since $I_n$ are nested, for any positive hyperintegers $m$ and $n$ we have $a_m \le a_{m+n} \le b_{m+n} \le b_n$, so that $\xi \le b_n$ for each $n \in \mathbb{N}^\#$. Since we obviously have $a_n \le \xi$ for each $n \in \mathbb{N}^\#$, we have $a_n \le \xi \le b_n$ for all $n \in \mathbb{N}^\#$, which implies $\xi \in \cap_{n=1}^\infty I_n$. Finally, if $\xi, \eta \in \cap_{n=1}^\infty I_n$, with $\xi \le \eta$, then we get $0 \le \eta - \xi \le b_n - a_n$, for all $n \in \mathbb{N}^\#$, so that $0 \le \eta - \xi \le \inf_{n \in \mathbb{N}^\#} |b_n - a_n| = 0$.

**Theorem 10.12. (Generalized Squeeze Theorem)**

Let $\{a_n\}$, $\{c_n\}$ be two hyper infinite sequences #-converging to $L$, and $\{b_n\}$ a hyper infinite sequence. If $\forall n \ge K, K \in \mathbb{N}^\#$ we have $a_n \le b_n \le c_n$, then $\{b_n\}$ also #-converges to $L$.

**Proof.** Choose an $\varepsilon > 0$, $\varepsilon \approx 0$. By definition of the #-limit, there is an $N_1 \in \mathbb{N}^\#$ such that for all $n > N_1$ we have $|a_n - L| < \varepsilon$, in other words $L - \varepsilon < a_n < L + \varepsilon$. Similarly, there is an $N_2 \in \mathbb{N}^\#$ such that for all $n > N_2$ we have $L - \varepsilon < c_n < L + \varepsilon$. Denote $N = \max(N_1, N_2, K)$. Then for $n > N, L - \varepsilon < a_n \le b_n \le c_n < L + \varepsilon$, in other words $|b_n - L| < \varepsilon$. Since $\varepsilon > 0, \varepsilon \approx 0$ was arbitrary, by definition of the #-limit this says that $\#\text{-lim}_{n \to \infty}^\# b_n = L$.

**Theorem 10.13. (Corollary of the Generalized Squeeze Theorem).**

If $\#\text{-lim}_{n \to \infty}^\# |a_n| = 0$ then $\#\text{-lim}_{n \to \infty}^\# a_n = 0$.

**Proof.** We know that $-|a_n| \le a_n \le |a_n|$. We want to apply the Generalized Squeeze Theorem. We are given that $\#\text{-lim}_{n \to \infty}^\# |a_n| = 0$. This also implies that $\#\text{-lim}_{n \to \infty}^\# (-|a_n|) = 0$. So by the Generalized Squeeze Theorem, $\#\text{-lim}_{n \to \infty}^\# a_n = 0$.

**Theorem 10.14. (Generalized Bolzano-Weierstrass Theorem)**

Every hyperbounded hyper infinite $\mathbb{R}_c^\#$-valued sequence has a #-convergent hyper infinite subsequence.

**Proof.** Let $\{w_n\}_{n \in \mathbb{N}^\#}$ be a hyperbounded hyper infinite sequence. Then, there exists an interval $[a_1, b_1]$ such that $a_1 \le w_n \le b_1$ for all $n \in \mathbb{N}^\#$.

Either $[a_1, \frac{a_1+b_1}{2}]$ or $[\frac{a_1+b_1}{2}, b_1]$ contains hyper infinitely many terms of $\{w_n\}$. That is, there exists hyper infinitely many $n$ in $\mathbb{N}^\#$ such that $a_n$ is in $[a_1, \frac{a_1+b_1}{2}]$ or there exists hyper infinitely many $n$ in $\mathbb{N}^\#$ such that $a_n$ is in $[\frac{a_1+b_1}{2}, b_1]$. If $[a_1, \frac{a_1+b_1}{2}]$ contains hyper infinitely many terms of $\{w_n\}$, let $[a_2, b_2] = [a_1, \frac{a_1+b_1}{2}]$. Otherwise, let $[a_2, b_2] = [\frac{a_1+b_1}{2}, b_1]$.

Either $[a_2, \frac{a_2+b_2}{2}]$ or $[\frac{a_2+b_2}{2}, b_2]$ contains hyper infinitely many terms of $\{w_n\}_{n \in \mathbb{N}^\#}$. If $[a_2, \frac{a_2+b_2}{2}]$ contains hyper infinitely many terms of $\{w_n\}$, let $[a_3, b_3] = [a_2, \frac{a_2+b_2}{2}]$. Otherwise, let $[a_3, b_3] = [\frac{a_2+b_2}{2}, b_2]$. By hyper infinite induction, we can continue this construction and obtain hyper infinite sequence of intervals $\{[a_n, b_n]\}_{n \in \mathbb{N}^\#}$ such that:

(i) for each $n \in \mathbb{N}^\#$, $[a_n, b_n]$ contains hyper infinitely many terms of $\{w_n\}_{n \in \mathbb{N}^\#}$,

(ii) for each $n \in \mathbb{N}^\#$, $[a_{n+1}, b_{n+1}] \subseteq [a_n, b_n]$ and

(iii) for each $n \in \mathbb{N}^\#$, $b_{n+1} - a_{n+1} = \frac{1}{2}(b_n - a_n)$.
Then generalized nested intervals theorem implies that the intersection of all of the intervals $[a_n, b_n]$ is a single point $w$. We will now construct a hyper infinite subsequence of $\{w_n\}_{n \in \mathbb{N}^\#}$ which will #-converge to $w$.

Since $[a_1, b_1]$ contains hyper infinitely many terms of $\{w_n\}_{n \in \mathbb{N}^\#}$, there exists $k_1 \in \mathbb{N}^\#$ such that $w_{k_1}$ is in $[a_1, b_1]$. Since $[a_2, b_2]$ contains hyper infinitely many terms of $\{w_n\}_{n \in \mathbb{N}^\#}$, there exists $k_2 \in \mathbb{N}^\#$, $k_2 > k_1$, such that $w_{k_2}$ is in $[a_2, b_2]$. Since $[a_3, b_3]$ contains hyper infinitely many terms of $\{w_n\}_{n \in \mathbb{N}^\#}$, there exists $k_3 \in \mathbb{N}^\#$, $k_3 > k_2$, such that $w_{k_3}$ is in $[a_3, b_3]$. Continuing this process by hyper infinite induction, we obtain hyper infinite sequence $\{w_{k_n}\}_{n \in \mathbb{N}^\#}$ such that $w_{k_n} \in [a_n, b_n]$ for each $n \in \mathbb{N}^\#$. The sequence $\{w_{k_n}\}_{n \in \mathbb{N}^\#}$ is a subsequence of $\{w_n\}_{n \in \mathbb{N}^\#}$ since $k_{n+1} > k_n$ for each $n \in \mathbb{N}^\#$. Since $a_n \to_{\#} w$, and $a_n \le w_n \le b_n$ for each $n \in \mathbb{N}^\#$, the squeeze theorem implies that $w_{k_n} \to_{\#} w$.

**Definition 10.12.** Let $\{a_n\}$ be a hyperreal sequence i.e., $a_n \in *\mathbb{R}_c^\#$, $n \in \mathbb{N}^\#$. Say that $\{a_n\}$ #-tends to 0 if, given any $\varepsilon > 0$, $\varepsilon \approx 0$, there is a hypernatural number $N \in \mathbb{N}^\#\mathbb{N}$, $N = N(\varepsilon)$ such that, for all $n > N$, $|a_n| \le \varepsilon$. We often denote this symbolically by $a_n \to_{\#} 0$. We can also, at this point, define what it means for a hyperreal sequence #-tends to a given number $q \in *\mathbb{R}_c^\# : \{a_n\}$ #-tends to $q$ if the hyperreal sequence $\{a_n - q\}$ #-tends to 0 i.e., $a_n - q \to_{\#} 0$.

**Definition 10.13.** Let $\{a_n\}$, $n \in \mathbb{N}^\#$ be a hyperreal sequence. We call $\{a_n\}$ a Cauchy hyperreal sequence if the difference between its terms #-tends to 0. To be precise: given any hyperreal number $\varepsilon > 0$, $\varepsilon \approx 0$, there is a hypernatural number $N = N(\varepsilon)$ such that for any $m, n > N$, $|a_n - a_m| < \varepsilon$.

**Theorem 10.15.** If $\{a_n\}$ is a #-convergent hyperreal sequence (that is, $a_n \to_{\#} b$ for some hyperreal number $b \in \mathbb{R}_c^\#$), then $\{a_n\}$ is a Cauchy hyperreal sequence.

**Theorem 10.16.** If $\{a_n\}$ is a Cauchy hyperreal sequence, then it is hyper bounded; that is, there is some $M \in \mathbb{R}_c^\#$ such that $|a_n| \le M$ for all $n \in \mathbb{N}^\#$.

**Theorem 10.17.** Any Cauchy hyperreal sequence $\{a_n\}$ has a #-limit in $*\mathbb{R}_c^\#$ i.e., there exists $b \in *\mathbb{R}_c^\#$ such that $a_n \to_{\#} b$.

**Proof.** By Definition 10.13 given $\varepsilon > 0$, $\varepsilon \approx 0$, there is a hypernatural number $N = N(\varepsilon)$ such that for any $n, n' > N$,

$$|a_n - a_{n'}| < \varepsilon. \qquad (10.1)$$

From (10.1) for any $n, n' > N$ we get

$$a_{n'} - \varepsilon < a_n < a_n + \varepsilon. \qquad (10.2)$$

The generalized Bolzano-Weierstrass theorem implies there is a #-convergent hyper infinite subsequence $\{a_{n_k}\} \subset \{a_n\}$ such that $a_{n_k} \to_{\#} b$ for some hyperreal number $b \in *\mathbb{R}_c^\#$. Let us show that the sequence $\{a_n\}$ also #-convergent to this $b \in *\mathbb{R}_c^\#$.

We can choose $k \in \mathbb{N}^\#$ so large that $n_k > N$ and

$$|a_{n_k} - b| < \varepsilon. \qquad (10.3)$$

We choose now in (10.1) $n' = n_k$ and therefore

$$|a_n - a_{n_k}| < \varepsilon. \qquad (10.4)$$

From (10.3) and (10.4) for any $n > N$ we get

$$(a_{n_k} - b) + (a_n - a_{n_k}) = |a_n - b| < 2\varepsilon. \qquad (10.5)$$

Thus $a_n \to_{\#} b$ as well.
**Remark 10.3.** Note that there exist canonical natural embedings

$$ \mathbb{R} \hookrightarrow *\mathbb{R} \hookrightarrow *\mathbb{R}_c^\# \qquad (10.6) $$

## 10.2. The Extended Hyperreal Number System *$\hat{\mathbb{R}}_c^\#$*

**Definition 10.14.** (a) A set $S \subset \mathbb{N}^\#$ is hyperfinite if $card(S) = card(\{x|0 \le x \le n\})$, $n \in \mathbb{N}^\# \setminus \mathbb{N}$. (b) A set $S \subset \mathbb{N}^\#$ is hyper infinite if $card(S) = card(\mathbb{N}^\#)$.

**Notation 10.2.** If $F$ is an arbitrary collection of subsets of $*\mathbb{R}_c^\#$, then $\cup\{S|S \in F\}$ is the set of all elements that are members of at least one of the sets in $F$, and $\cap\{S|S \in F\}$ is the set of all elements that are members of every set in $F$. The union and intersection of finitely or hyperfinitely many sets $S_k, 0 \le k \le n \in \mathbb{N}^\#$ are also written as $\cup_{k=0}^n S_k$ and $\cap_{k=0}^n S_k$. The union and intersection of an hyperinfinite sequence $S_k, k \in \mathbb{N}^\#$ of sets are written as $\cup_{k=0}^\infty S$ or $\cup_{n \in \mathbb{N}^\#} S$ and $\cap_{k=0}^\infty S$ or $\cap_{n \in \mathbb{N}^\#} S$ correspondingly. A nonempty set $S$ of hyperreal numbers $*\mathbb{R}_c^\#$ is unbounded above if it has no hyperfinite

upper bound, or unbounded below if it has no hyperfinite lower bound. It is convenient to adjoin to the hyperreal number system two points, $+\infty^\#$ (which we also write more simply as $\infty^\#$) and $-\infty^\#$, and to define the order relationships between them and any hyperreal number $x \in *\mathbb{R}_c^\#$ by $-\infty^\# < x < \infty^\#$.

We call $-\infty^\#$ and $\infty^\#$ points at hyperinfinity. If $S$ is a nonempty set of hyperreals, we write $sup_S = \infty^\#$ to indicate that $S$ is unbounded above, and $inf_S = -\infty^\#$ to indicate that $S$ is unbounded below.

## #-Open and #-Closed Sets on *$\hat{\mathbb{R}}_c^\#$*

**Definition 10.15.** If $a$ and $b$ are in the extended hyperreals and $a < b$, then the open interval $(a, b)$ is defined by $(a, b) \triangleq \{x|a < x < b\}$. :

The open intervals $(a, +\infty^\#)$ and $(-\infty^\#, b)$ are semi-hyperinfinite if $a$ and $b$ are finite or hyperfinite, and $(-\infty^\#, \infty^\#)$ is the entire hyperreal line.

If $-\infty^\# < a < b < \infty^\#$, the set $[a, b] \triangleq \{x|a \le x \le b\}$ is #-closed, since its complement is the union of the #-open sets $(-\infty^\#, a)$ and $(b, \infty^\#)$. We say that $[a, b]$ is a #-closed interval. Semi-hyper infinite #-closed intervals are sets of the form $[a, \infty) = \{x|a \le x\}$ and $(-\infty^\#, a] = \{x|x \le a\}$, where $a$ is finite or hyperfinite. They are #-closed sets, since their complements are the #-open intervals $(-\infty^\#, a)$ and $(a, \infty^\#)$, respectively.

**Definition 10.16.** If $x_0 \in \mathbb{R}_c^\#$ is a hyperreal number and $\varepsilon > 0, \varepsilon \approx 0$ then the open interval

$(x_0 - \varepsilon, x_0 + \varepsilon)$ is an #-neighborhood of $x_0$. If a set $S \subset *\mathbb{R}_c^\#$ contains an #-neighborhood of $x_0$, then $S$ is a #-neighborhood of $x_0$, and $x_0$ is an #-interior point of $S$.

The set of #-interior points of $S$ is the #-interior of $S$, denoted by #-Int($S$).

(i) If every point of $S$ is an #-interior point (that is, $S = #-Int(S)$), then $S$ is #-open.

(ii) A set $S$ is #-closed if $S^c = *\mathbb{R}_c^\# \setminus S$ is #-open.

**Example 10.1.** An open interval $(a, b)$ is an #-open set, because if $x_0 \in (a, b)$ and $\varepsilon \le \min\{x_0 - a; b - x_0\}$, then $(x_0 - \varepsilon, x_0 + \varepsilon) \subset (a, b)$

**Remark 10.4.** The entire hyperline $*\hat{\mathbb{R}}_c^\# = (-\infty^\#, \infty^\#)$ is #-open, and therefore $\emptyset$ is
#-closed.

However, $\emptyset$ is also #-open, for to deny this is to say that $\emptyset$ contains a point that is not an #-interior point, which is absurd because $\emptyset$ contains no points. Since $\emptyset$ is #-open, $\hat{\mathbb{R}}_c^\#$ is #-closed. Thus, $\hat{\mathbb{R}}_c^\#$ and $\emptyset$ are both #-open and #-closed.

**Remark 10.5.** They are not the only subsets of $\hat{\mathbb{R}}_c^\#$ with this property.

**Definition 10.17.** A deleted #-neighborhood of a point $x_0$ is a set that contains every point

of some #-neighborhood of $x_0$ except for $x_0$ itself. For example, $S = \{x|0 < |x - x_0| < \varepsilon\}$, where $\varepsilon \approx 0$, is a deleted #-neighborhood of $x_0$. We also say that it is a deleted $\varepsilon$-#-neighborhood of $x_0$.

**Theorem 10.18.** (a) The union of #-open sets is #-open:

(b) The #-intersection of #-closed sets is #-closed:

These statements apply to arbitrary collections, hyperfinite or hyperinfinite, of #-open and #-closed sets.

**Proof** (a) Let $L$ be a collection of #-open sets and $S = \bigcup \{G | G \in L\}$.

If $x_0 \in S$, then $x_0 \in G_0$ for some $G_0$ in $L$, and since $G_0$ is #-open, it contains some $\varepsilon$-#-neighborhood of $x_0$. Since $G_0 \subset S$, this $\varepsilon$-#-neighborhood is in $S$, which is consequently a #-neighborhood of $x_0$. Thus, $S$ is a #-neighborhood of each of its points, and therefore #-open, by definition.

(b) Let $F$ be a collection of #-closed sets and $T = \bigcap \{H | H \in F\}$. Then $T^c = \bigcup \{H^c | H \in F\}$ and, since each $H^c$ is #-open, $T^c$ is #-open, from (a). Therefore, $T$ is #-closed, by definition.

**Example 10.2.** If $-\infty^\# < a < b < \infty^\#$, the set $[a, b] = \{x|a \le x \le b\}$ is #-closed, since its complement is the union of the #-open sets $(-\infty^\#a)$ and $(b, \infty^\#)$. We say that $[a, b]$ is a #-closed interval. The set $[a, b) = \{x|a \le x < b\}$ is a half-#-closed or half-#-open interval if $-\infty^\# < a < b < \infty^\#$, as is $(a, b] = \{x|a < x \le b\}$ however, neither of these sets is #-open or #-closed. Semi-infinite #-closed intervals are sets of the form

$[a, \infty^\#) = \{x|a \le x\}$ and $(-\infty^\#, a] = \{x|x \le a\}$, where $a$ is hyperfinite. They are #-closed sets, since their complements are the #-open intervals $(-\infty^\#, a)$ and

$(a, \infty^\#)$, respectively.

**Definition 10.18.** Let $S$ be a subset of $\hat{\mathbb{R}}_c^\# = (-\infty^\#, \infty^\#)$. Then

(a) $x_0$ is a #-limit point of $S$ if every deleted #-neighborhood of $x_0$ contains a point of $S$.

(b) $x_0$ is a boundary point of $S$ if every #-neighborhood of $x_0$ contains at least one point in $S$ and one not in $S$. The set of #-boundary points of $S$ is the #-boundary of $S$, denoted

by $\#-\partial S$. The #-closure of $S$, denoted by $\#-\bar{S}$, is $S \cup \#-\partial S$.

(c) $x_0$ is an #-isolated point of $S$ if $x_0 \in S$ and there is a #-neighborhood of $x_0$ that contains

no other point of $S$.

(d) $x_0$ is #-exterior to $S$ if $x_0$ is in the #-interior of $S^c$. The collection of such points is the #-exterior of $S$.

**Theorem 10.19.** A set $S$ is #-closed if and only if no point of $S^c$ is a #-limit point of $S$.

**Proof.** Suppose that $S$ is #-closed and $x_0 \in S^c$. Since $S^c$ is #-open, there is a #-neighborhood of $x_0$ that is contained in $S^c$ and therefore contains no points of $S$.

Hence, $x_0$ cannot be a #-limit point of $S$. For the converse, if no point of $S^c$ is a #-limit point of $S$ then every point in $S^c$ must have a #-neighborhood contained
in $S^c$. Therefore, $S^c$ is #-open and $S$ is #-closed.

**Corollary 10.1.** A set $S$ is #-closed if and only if it contains all its #-limit points. If $S$ is #-closed and hyper bounded, then $\inf(S)$ and $\sup(S)$ are both in $S$.

**Proposition 10.1.** If $S$ is #-closed and hyper bounded, then $\inf(S)$ and $\sup(S)$ are both in $S$.

#-Open Coverings

**Definition 10.19.** A collection $H$ of #-open sets of $\mathbb{R}_c^\#$ is an #-open covering of a set $S$ if every point in $S$ is contained in a set $H$ belonging to $H$; that is, if $S \subset \cup\{F|F \in H\}$.

**Definition 10.20.** A set $S \subset \mathbb{R}_c^\#$ is called #-compact (or hyper compact) if each of its #-open covers has a hyperfinite subcover.

**Theorem 10.20. (Generalized Heine–Borel Theorem)** If $H$ is an #-open covering of a #-closed and hyper bounded subset $S$ of the hyperreal line $\mathbb{R}_c^\#$ (or of the $\mathbb{R}_c^{\#n}$, $n \in \mathbb{N}^\#$) then $S$ has an #-open

covering $\tilde{H}$ consisting of hyper finite many #-open sets belonging to $H$.

**Proof.** If a set $S$ in $\mathbb{R}_c^{\#n}$ is hyper bounded, then it can be enclosed within an $n$-box $T_0 = [-a, a]^n$ where $a > 0$. By the property above, it is enough to show that $T_0$ is #-compact.

Assume, by way of contradiction, that $T_0$ is not #-compact. Then there exists an hyper infinite open cover $C_{\infty\#}$ of $T_0$ that does not admit any hyperfinite subcover. Through bisection of each of the sides of $T_0$, the box $T_0$ can be broken up into $2n$ sub $n$-boxes, each of which has diameter equal to half the diameter of $T_0$. Then at least one of the $2n$ sections of $T_0$ must require an hyper infinite subcover of $C_{\infty\#}$, otherwise $C_{\infty\#}$ itself would have a hyperfinite subcover, by uniting together the hyperfinite covers of the sections. Call this section $T_1$. Likewise, the sides of $T_1$ can be bisected, yielding $2n$ sections of $T_1$, at least one of which must require an hyper infinite subcover of $C_{\infty\#}$.

Continuing in like manner yields a decreasing hyper infinite sequence of nested $n$-boxes: $T_0 \supset T_1 \supset T_2 \supset \dots \supset T_k \supset \dots$, $k \in \mathbb{N}^\#$, where the side length of $T_k$ is $(2a)/2^k$, which #-converges to 0 as $k$ tends to hyper infinity, $k \to \infty^\#$. Let us define a hyper infinite sequence $\{x_k\}_{k \in \mathbb{N}^\#}$ such that each $x_k : x_k \in T_k$. This hyper infinite sequence is Cauchy, so it must #-converge to some #-limit $L$. Since each $T_k$ is #-closed, and for each $k$ the sequence $\{x_k\}_{k \in \mathbb{N}^\#}$ is eventually always inside $T_k$, we see that $L \in T_k$ for each $k \in \mathbb{N}^\#$. Since $C_{\infty\#}$ covers $T_0$, then it has some member $U \in C_{\infty\#}$ such that $L \in U$. Since $U$ is open, there is an $n$-ball $B(L) \subseteq U$. For large enough $k$, one has $T_k \subseteq B(L) \subseteq U$, but then the infinite number of members of $C_{\infty\#}$ needed to cover $T_k$ can be replaced by just one: $U$, a contradiction. Thus, $T_0$ is #-compact. Since $S$ is #-closed and a subset of the #-compact set $T_0$, then $S$ is also #-compact.

As an application of the Generalized Heine–Borel theorem, we give a short proof of the Generalized Bolzano–Weierstrass Theorem.

**Theorem 10.21. (Generalized Bolzano–Weierstrass Theorem)** Every hyper bounded hyper infinite set $S \subset \mathbb{R}_c^\#$ has at least one #-limit point.

**Proof.** We will show that a hyper bounded nonempty set without a #-limit point can contain only finite or a hyper finite number of points. If $S$ has no #-limit points, then $S$ is #-closed (Theorem 9.) and every point $x \in S$ has an w-#-open neighborhood $N_x$ that contains no point of $S$ other than $x$. The collection $H = \{N_x|x \in S\}$ is an w-#-open
covering for $S$. Since $S$ is also hyper bounded, Theorem 9.3 implies that $S$ can be
covered by finite or a hyper finite collection of sets from $H$, say $N_{x_1}, \dots, N_{x_n}, n \in \mathbb{N}^#$.
Since these sets contain only $x_1, \dots, x_n$ from $S$, it follows that $S = \{x_k\}_{1 \le k \le n}, n \in \mathbb{N}^#$.

10.3. External Cauchy hyperreals $\mathbb{R}^#_c$ and $*\mathbb{R}^#_c$ axiomatically.

A model for the Cauchy hyperreal number system consists of a set $\mathbb{R}^#_c$, two distinct elements 0 and 1 of $\mathbb{R}^#_c$, two binary operations $+$ and $\times$ on $\mathbb{R}^#_c$ (called addition and multiplication, respectively), and a binary relation $\leq$ on $\mathbb{R}^#_c$, satisfying the following properties.

Axioms:

I.($\mathbb{R}^#_c$, $+$, $\times$) forms a field i.e.,

(i) For all $x, y$, and $z$ in $\mathbb{R}^#$, $x + (y + z) = (x + y) + z$ and $x \times (y \times z) = (x \times y) \times z$.

(associativity of addition and multiplication)

(ii) For all $x$ and $y$ in $\mathbb{R}^#$, $x + y = y + x$ and $x \times y = y \times x$.

(commutativity of addition and multiplication)

(iii) For all $x, y$, and $z$ in $\mathbb{R}^#$, $x \times (y + z) = (x \times y) + (x \times z)$.

(distributivity of multiplication over addition)

(iv) For all $x$ in $\mathbb{R}^#$, $x + 0 = x$.

(existence of additive identity)

0 is not equal to 1, and for all $x$ in $\mathbb{R}^#$, $x \times 1 = x$.

(existence of multiplicative identity)

(v) For every $x$ in $\mathbb{R}^#$, there exists an element $-x$ in $\mathbb{R}^#$, such that $x + (-x) = 0$.

(existence of additive inverses)

(vi) For every $x \neq 0$ in $\mathbb{R}^#$, there exists an element $x - 1$ in $\mathbb{R}^#$, such that $x \times x - 1 = 1$.

(existence of multiplicative inverses)

II. ($\mathbb{R}^#$, $\leq$) forms a totally ordered set. In other words,

(i) For all $x$ in $\mathbb{R}^#$, $x \leq x$. (reflexivity)

(ii) For all $x$ and $y$ in $\mathbb{R}^#$, if $x \leq y$ and $y \leq x$, then $x = y$. (antisymmetry)

(iii) For all $x, y$, and $z$ in $\mathbb{R}^#$, if $x \leq y$ and $y \leq z$, then $x \leq z$. (transitivity)

(iv) For all $x$ and $y$ in $\mathbb{R}^#$, $x \leq y$ or $y \leq x$. (totality)

The field operations $+$ and $\times$ on $\mathbb{R}^#$ are compatible with the order $\leq$. In other words,

(v) For all $x, y$ and $z$ in $\mathbb{R}^#$, if $x \leq y$, then $x + z \leq y + z$. (preservation of order under addition)

(vi) For all $x$ and $y$ in $\mathbb{R}^#$, if $0 \leq x$ and $0 \leq y$, then $0 \leq x \times y$ (preservation of order under multiplication)

III. Non-Archimedean property

$\mathbb{Q}^# \subset \mathbb{R}^# \text{ i.e., } \mathbb{R}^# \text{ is non-Archimedean ordered field.}$

**Remark 10.1.** Here a hyperrational is by definition a ratio of two hyperintegers.

Consider

the ring $\mathbb{Q}^#_{\text{fin}}$ of all limited (i.e. finite) elements in $\mathbb{Q}^#$. Then $\mathbb{Q}^#_{\text{fin}}$ has a unique maximal ideal $\mathbb{I}^#_{\approx}$, the infinitesimals or infinitesimal numbers are quantities that are closer to zero

than any real number from the field $\mathbb{R}$, but are not zero. The quotient ring $\mathbb{Q}^#_{\text{fin}}/\mathbb{I}^#_{\approx}$ gives the field $\mathbb{R}$ of real numbers.
**Definition 10.1.** An element $x \in \mathbb{R}^\#$ is called finite if $|x| < r$ for some $r \in \mathbb{Q}$, $r > 0$. As we shall see in a moment in bivalent case,

**Theorem 10.1.** Every finite $x \in \mathbb{R}^\#$ is infinitely close to some (unique) $r \in \mathbb{R}$ in the sense that $|x - r|$ is either 0 or positively infinitesimal in $\mathbb{R}^\#$. This unique $r$ is called the standard part of $x$ and is denoted by $st(x)$.

**Proof.** Let $x \in \mathbb{R}^\#$ be finite. Let $D_1$, be the set of $r \in \mathbb{R}$ such that $r < x$ and $D_2$ the set of $r' \in \mathbb{R}$ such that $x < r'$. The pair $(D_1, D_2)$ forms a Dedekind cut in $\mathbb{R}$, hence determines a unique $r_0 \in \mathbb{R}$. A simple argument shows that $|x - r_0|$ is infinitesimal, i.e., $st(x) = r_0$. 

**Notation 10.1.** We usually write $x \approx 0$ iff $x \in \mathbb{I}^\#_\approx$.

**Definition 10.2.** A hypersequence of hyperreal numbers is any function $a : \mathbb{N}^\# \to \mathbb{R}^\#$. Often hypersequences such as these are called hyperreal hypersequences, hypersequences of hyperreal numbers or hypersequences in $\mathbb{R}^\#$ to make it clear that the elements of the sequence are hyperreal numbers. Analogous definitions can be given for sequences of hypernatural numbers, hyperintegers, etc.

**Notation 10.2.** However, we usually write $a_n$ for the image of $n \in \mathbb{N}^\#$ under $a$, rather than $a(n)$. The values $a_n$ are often called the elements of the hypersequence $(x_n)_{n \in \mathbb{N}^\#}$.

**Definition 10.3.** We call $x \in \mathbb{R}^\#$ the limit of the hypersequence $(x_n)_{n \in \mathbb{N}^\#}$ if the following condition holds: for each hyperreal number $\varepsilon \in \mathbb{R}^\#$ such that $\varepsilon \approx 0$, $\varepsilon > 0$, there exists a hypernatural number $N \in \mathbb{N}^\#$ such that, for every hypernatural number $n \ge N$, we have $|x_n - x| < \varepsilon$.

**Definition 10.4.** The hypersequence $(x_n)_{n \in \mathbb{N}^\#}$ is said to #-converge to the #-limit $x$, written $x_n \to x, n \to \infty^\#$ or $\lim_{n \to \infty^\#} (x_n) = x$. Symbolically, this reads:

$$ \forall \varepsilon [(\varepsilon \approx 0) \land (\varepsilon > 0)][\exists N \in \mathbb{N}^\# (\forall n \in \mathbb{N}^\# (n \ge N \Rightarrow |x_n - x| < \varepsilon))]. \quad (10.1) $$

If a hypersequence $(x_n)_{n \in \mathbb{N}^\#}$ converges to some limit, then it is convergent; otherwise it is #-divergent. A hypersequence that has zero as a #-limit is sometimes called a null hypersequence.

Limits of hypersequences behave well with respect to the usual arithmetic operations.

If $a_n \to a, n \to \infty^\#$ and $b_n \to b, n \to \infty^\#$, then $a_n + b_n \to a + b, n \to \infty^\#$ and $a_n \times b_n \to a \times b, n \to \infty^\#$ if neither $b_n$ or any $b_n$ is zero, $a_n \times b_n \to a \times b, n \to \infty^\#$.

The following properties of limits of real hypersequences provided, in each equation below, that the limits on the right exist.

The limit of a hypersequence is unique.

1. \#-lim$_{n \to \infty^\#}$ $(a_n \pm b_n) = \#-\lim_{n \to \infty^\#} a_n \pm \#-\lim_{n \to \infty^\#} b_n$

2. \#-lim$_{n \to \infty^\#}$ $(c \times a_n) = c \times \#-\lim_{n \to \infty^\#} a_n$

3. \#-lim$_{n \to \infty^\#}$ $(a_n \times b_n) = (\#-\lim_{n \to \infty^\#}) \times (\#-\lim_{n \to \infty^\#} b_n)$

4. \#-lim$_{n \to \infty^\#}$ $(a_n/b_n) = \#-\lim_{n \to \infty^\#} a_n/\#-\lim_{n \to \infty^\#} b_n \text{ provided } \#-\lim_{n \to \infty^\#} b_n \neq 0$

5. \#-lim$_{n \to \infty^\#}$ $a_n^p = [\#-\lim_{n \to \infty^\#} a_n]^p$

6. If $a_n \le b_n$ where $n$ greater than some $N$, then $\#-\lim_{n \to \infty^\#} a_n \le \#-\lim_{n \to \infty^\#} b_n$

7. (Squeeze theorem) If $a_n \le c_n \le b_n$, and $\#-\lim_{n \to \infty^\#} a_n = \#-\lim_{n \to \infty^\#} b_n = L$, then
$\#\text{-lim}_{n\to\infty} c_n = L$.

**Definition 10.5.** A hyper infinite sequence $(x_n)$ is said to tend to hyperinfinity, written $x_n \to \infty^\# $ or $\#\text{-lim}_{n\to\infty} x_n = \infty^\# $, if for every $K \in \mathbb{R}^\#$, there is an $N \in \mathbb{N}^\#$ such that for every $n \ge N$; that is, the hypersequence terms are eventually larger than any fixed $K$.

Similarly, $x_n \to -\infty^\#$ if for every $K \in \mathbb{R}^\#$, there is an $N \in \mathbb{N}^\#$ such that for every $n \ge N$, $x_n < K$. If a hypersequence tends to infinity or minus infinity, then it is divergent.

However, a divergent hypersequence need not tend to plus or minus hyperinfinity

**Definition 10.6.** A hypersequence $(x_n)_{n\in\mathbb{N}^\#}$ of hyperreal numbers is called a *Cauchy hypersequence* if for every positive hyperreal number $\epsilon$, there is a positive

hyperinteger

$N \in \mathbb{N}^\#$ such that for all hypernatural numbers $m, n > N$ : $|x_m - x_n| < \epsilon$, where the vertical bars denote the absolute value. In a similar way one can define

Cauchy hypersequences

of hyperrational numbers, etc. Cauchy formulated such a condition by requiring $|x_m - x_n| \approx 0$ i.e., to be infinite small for every pair of infinite large $m, n \in \mathbb{N}^\#$.

**Definition 10.7.** Let $\mathbb{R}_c^\#$ be the set of Cauchy hypersequences of hyperrational numbers.

That is, hypersequences $(x_n)_{n\in\mathbb{N}^\#}$ of hyperrational numbers such that for every hyperrational $\epsilon > 0$, there exists an hyperinteger $N \in \mathbb{N}^\#\mathbb{N}$ such that for all hypernatural numbers $m, n > N$, $|x_m - x_n| < \epsilon$. Here the vertical bars as usual denote the absolute value.

**Definition 10.8.** A standard procedure to force all Cauchy hypersequences in a metric space to converge is adding new points to the metric space in a process called completion. $\mathbb{R}_c^\#$ is defined as the completion of $\mathbb{Q}^\#$ with respect to the metric $|x - y|$, as will be detailed below.

**Definition 10.9.** Cauchy hypersequences $(x_n)_{n\in\mathbb{N}^\#}$ and $(y_n)_{n\in\mathbb{N}^\#}$ can be added and multiplied as follows:

$$ (x_n)_{n \in \mathbb{N}^\#} + (y_n)_{n \in \mathbb{N}^\#} = (x_n + y_n)_{n \in \mathbb{N}^\#}, \qquad (10.2) $$

and

$$ (x_n)_{n \in \mathbb{N}^\#} \times (y_n)_{n \in \mathbb{N}^\#} = (x_n \times y_n)_{n \in \mathbb{N}^\#}. \qquad (10.3) $$

**Definition 10.10.** Two Cauchy hypersequences are called equivalent if and only if the difference between them tends to zero. This defines an equivalence relation that is compatible with the operations (10.2)-(10.3) defined above, and the set $\mathbb{R}_c^\#$ of all equivalence classes $cl[(x_n)_{n\in\mathbb{N}^\#}]$ can be shown to satisfy all axioms of the hyperreal numbers.

We can embed $\mathbb{Q}^\#$ into $\mathbb{R}_c^\#$ by identifying the rational number $r \in \mathbb{Q}^\#$ with the equivalence

class of the hypersequence $(r_n)_{n\in\mathbb{N}^\#}$ with $r_n = r$ for all $n \in \mathbb{N}^\#$.

**Remark 10.2.** Comparison between hyperreal numbers is obtained by defining the following comparison between Cauchy hypersequences:

$$ (x_n)_{n \in \mathbb{N}^\#} \ge (y_n)_{n \in \mathbb{N}^\#} \qquad (10.4) $$

if and only if $x$ is equivalent to $y$ or there exists an hyperinteger $N \in \mathbb{N}^\#$ such that

$x_n \ge y_n$

for all $n > N$.
**Remark 10.3.** By construction, every hyperreal number $x \in \mathbb{R}_c^\#$ is represented by a Cauchy

hypersequence of hyperrational numbers. This representation is far from unique; every

hyperrational hypersequence that converges to $x$ is a representation of $x$. This reflects the observation that one can often use different hypersequences to approximate the same hyperreal number. The equation $0.999... = 1$ states that the hypersequences $(0, 0.9, 0.99, 0.999,...)$ and $(1, 1, 1, 1,...)$ are equivalent, i.e., their difference #-converges

to 0.

**IV.** The field $\mathbb{R}^\#$ is complete in the following sense:

**Definition 10.11.** Let $S \subsetneq \mathbb{R}_c^\#$ be a non-empty set of hyperreal numbers.

A hyperreal number $x \in \mathbb{R}_c^\#$ is called an upper bound for $S$ if $x \ge s$ for all $s \in S$.

A hyperreal number $x$ is the least upper bound (or supremum $\sup S$) for $S$ if $x$ is an upper

bound for $S$ and $x \le y$ for every upper bound $y$ of $S$.

**Remark 10.4.** The order $\le$ given by Eq. (3.4) obviously is $\le$-incomplete.

**Definition 10.12.** Let $S \subsetneq \mathbb{R}_c^\#$ be a nonempty subset of $\mathbb{R}_c^\#$. We will say that:

(1) $S$ is $\le$-admissible above if the following conditions are satisfied:

(i) $S$ bounded above;

(ii) let $A(S)$ be a set $\forall x[x \in A(S) \Leftrightarrow x \ge S]$ then for any $\varepsilon > 0, \varepsilon \approx 0$ there exist $\alpha \in S$ and $\beta \in A(S)$ such that $\beta - \alpha \le \varepsilon \approx 0$.

(2) $S$ is $\le$-admissible below if the following condition are satisfied:

(i) $S$ bounded below;

(ii) let $L(S)$ be a set $\forall x[x \in L(S) \Leftrightarrow x \le S]$ then for any $\varepsilon > 0, \varepsilon \approx 0$ there exist $\alpha \in S$ and $\beta \in L(S)$ such that $\alpha - \beta \le \varepsilon \approx 0$.

**Theorem 10.2.** (i) Every $\le$-admissible above subset $S \subsetneq \mathbb{R}_c^\#$ has a supremum $\sup S$.

(ii) Every $\le$-admissible below subset $S \subsetneq \mathbb{R}_c^\#$ has infinum $\inf S$.

**Proof.** Let $S \subsetneq \mathbb{R}_c^\#$ be a nonempty subset of $\mathbb{R}_c^\#$, and let $M \in \mathbb{Q}^\#$ be an hyperrational upper bound for $S$. We are going to construct two hypersequences of hyperrational numbers, $(u_n)_{n \in \mathbb{N}^\#$}$ and $(l_n)_{n \in \mathbb{N}^\#}$. First, since $S$ is nonempty, there is some element $s_0 \in S$.

We can choose a hyperrational number $L \in \mathbb{Q}^\#$ such that $L < s_0$. Now, we go through the following hyperinductive procedure to produce hyperrational numbers $u_0, u_1, u_2, ...$ and $l_0, l_1, l_2, l_3, ...$.

(i) Set $u_0 = M$ and $l_0 = L$.

(ii) Suppose that we have already defined $u_n$ and $l_n, n \in \mathbb{N}^\#$.

Consider the number $m_n = (u_n + l_n)/2$, i.e., the average between $u_n$ and $l_n$.

(1) If $m_n$ is an upper bound for $S$, define $u_{n+1} = m_n$ and $l_{n+1} = l_n$.

(2) If $m_n$ is not an upper bound for $S$, define $u_{n+1} = u_n$ and $l_{n+1} = m_n$.

Since $l_0 < M$, it is easy to prove by hyperinfinite induction that $(u_n)_{n \in \mathbb{N}^\#$}$ is a non-increasing hypersequence, i.e. $u_{n+1} \le u_n$ and $(l_n)_{n \in \mathbb{N}^\#}$ is a non-decreasing hypersequence, i.e. $l_{n+1} \ge l_n$.

**Remark 10.5.** Note that in the first case above, we have that

$$u_{n+1} - l_{n+1} = m_n - l_n = \frac{u_n + l_n}{2} - l_n = \frac{u_n - l_n}{2} \qquad (10.5)$$
In the second case, we also have that

$$u_{n+1} - l_{n+1} = u_n - m_n = u_n - \frac{u_n + l_n}{2} = \frac{u_n - l_n}{2}. \quad (10.6)$$

Now, this means that $u_1 - l_1 = \frac{1}{2}(M-L)$ and so $u_2 - l_2 = \frac{1}{2}(u_1 - l_1) = \frac{1}{2^2}(M-L)$, and in general by hyperinfinite induction one obtains

$$u_n - l_n = 2^{-n}(M - L). \quad (10.7)$$

Since $M > L$ so $M - L > 0$, and since $2^{-n} < n^{-1}$ we have for any $\varepsilon > 0$, $\varepsilon \approx 0$ that $2^{-n}(M - L) < \varepsilon$ for all sufficiently large $n \in \mathbb{N}^*\mathbb{N}$. Thus, $u_n - l_n < \varepsilon$ as well, and so

$$\#\lim_{n \to \infty} (u_n - l_n) = 0. \quad (10.8)$$

This defines two hypersequences of hyperrationals, and so we have hyperreal numbers

$l = (l_n)_{n \in \mathbb{N}^*}$ and $u = (u_n)_{n \in \mathbb{N}^*}$. It is easy to prove, by induction on $n \in \mathbb{N}^*$ that:

(i) $u_n$ is an upper bound for $S$ for all $n \in \mathbb{N}^*$ and

(ii) $l_n$ is never an upper bound for $S$ for any $n \in \mathbb{N}^*$.

Thus $u$ is an upper bound for $S$. To see that it is a least upper bound, notice that the \#-limit of $(u_n - l_n)_{n \in \mathbb{N}^*}$ is 0, and so $l = u$. Now suppose $b < u = l$ is a smaller upper bound

for $S$. Since $(l_n)_{n \in \mathbb{N}^*}$ is monotonic increasing it is easy to see that $b < l_n$ for some $n \in \mathbb{N}^*$.

But $l_n$ is not an upper bound for $S$ and so neither is $b$. Hence $u$ is a least upper bound for $S$.

# 11. Basic analysis on external non-Archimedean field $\mathbb{R}_c^\#$.

## 11.1. The \#-limit of a function $f: \mathbb{R}_c^\# \to \mathbb{R}_c^\#$

**Definition 11.1.** The $(\varepsilon, \delta)$ definition of the \#-limit of a function $f: D \to \mathbb{R}_c^\#$ is as follows: Let $f$ be a $\mathbb{R}_c^\#$-valued function defined on a subset $D \subset \mathbb{R}_c^\#$ of the Cauchy hyperreal numbers. Let $c$ be a \#-limit point of $D$ and let $L$ be a hyperreal number. We say that

$$\#\lim_{x \to c} f(x) = L \quad (11.1)$$

if for every $\varepsilon \approx 0, \varepsilon > 0$ there exists a $\delta \approx 0, \delta > 0$ such that, for all $x \in D$, if $0 < |x - c| < \delta$, then $|f(x) - L| < \varepsilon$, symbolically:

$$\lim_{x \to c} f(x) = L \iff (\forall \varepsilon > 0 \land \varepsilon \approx 0 \land \varepsilon > 0) \exists \delta > 0 \forall x \in D, 0 < |x - c| < \delta \Rightarrow |f(x) - L| < \varepsilon. \quad (11.2)$$

**Definition 11.2.** The function $f: \mathbb{R}_c^\# \to \mathbb{R}_c^\#$ is \#-continuous (or micro continuous) at some

point $c$ of its domain if the \#-limit of $f(x)$, as $x$ \#-approaches $c$ through the domain of $f$, exists and is equal to $f(c)$:

$$\#\lim_{x \to c} f(x) = f(c). \quad (11.3)$$

**Theorem 11.1.** If \#-lim$_{x \to c} x_0 f(x)$ exists; then it is unique that is; if \#-lim$_{x \to c} x_0 f(x) = L_1$ and \#-lim$_{x \to c} x_0 f(x) = L_2$, then $L_1 = L_2$.

**Theorem 11.2.** If \#-lim$_{x \to c} x_0 f_1(x) = L_1$ and \#-lim$_{x \to c} x_0 f_2(x) = L_2$ then
$$
\begin{align}
\#\text{-lim}_{x\to\#} x_0 [f_1(x) \pm f_2(x)] &= L_1 \pm L_2, \notag \\
\#\text{-lim}_{x\to\#} x_0 [f_1(x) \times f_2(x)] &= L_1 \times L_2, \notag \\
\#\text{-lim}_{x\to\#} x_0 \frac{f_1(x)}{f_2(x)} &= \frac{L_1}{L_2}, L_2 \neq 0. \tag{11.4}
\end{align}
$$

**Definition 11.3. (a)** We say that $f(x)$ #-approaches the left-hand #-limit $L$ as $x$ #-approaches $x_0$ from the left, and write $\#\text{-lim}_{x\to x_0^-} f(x) = L$, if $f(x)$ is defined on some #-open interval $(a, x_0)$ and, for each $\varepsilon > 0, \varepsilon \approx 0$ there is a $\delta > 0, \delta \approx 0$ such that $|f(x) - L| < \varepsilon$ if $x_0 - \delta < x < x_0$.

(b) We say that $f(x)$ #-approaches the right-hand #-limit $L$ as $x$ #-approaches $x_0$ from the right, and write $\#\text{-lim}_{x\to x_0^+} f(x) = L$, if $f(x)$ is defined on some open interval $(x_0, b)$ and, for each $\varepsilon > 0$, there is a $\delta > 0, \delta \approx 0$ such that $|f(x) - L| < \varepsilon, \varepsilon > 0, \varepsilon \approx 0$ if $x_0 < x < x_0 + \delta$. Left- and right-hand #-limits are also called one-sided #-limits. We will often simplify the notation by writing $\#\text{-lim}_{x\to x_0^-} f(x) = f(x_0^-)$ and $\#\text{-lim}_{x\to x_0^+} f(x) = f(x_0^+)$.

**Theorem 11.3.** A function $f$ has a #-limit at $x_0$ if and only if it has left- and right-hand #-limits at $x_0$; and they are equal. More specifically; $\#\text{-lim}_{x\to x_0} f(x) = L$ if and only if $f(x_0^+) = f(x_0^-) = L$.

**Definition 11.4.** We say that $f(x)$ approaches the #-limit $L$ as $x$ approaches $\infty^{\#}$, and write $\#\text{-lim}_{x\to\#} \infty^{\#} f(x) = L$, if $f$ is defined on an interval $(a, \infty^{\#})$ and, for each $\varepsilon > 0, \varepsilon \approx 0$, there is a number $\beta$ such that $|f(x)| = L| < \varepsilon$ if $x > \beta$.

**Definition 11.5.** We say that $f(x)$ approaches $\infty^{\#}$ as $x$ approaches $x_0$ from the left, and write

$$
\#\text{-lim}_{x\to\#} x_0^- f(x) = \infty^{\#} \text{ or } f(x_0^-) = \infty^{\#} \qquad (11.5)
$$

if $f$ is defined on an interval $(a, x_0)$ and, for each hyperreal number $M$, there is a $\delta \approx 0, \delta > 0$ such that $f(x) > M$ if $x_0 - \delta < x < x_0$.

Similarly we define: $\#\text{-lim}_{x\to x_0^-} f(x) = -\infty^{\#}$, $\#\text{-lim}_{x\to x_0^+} f(x) = -\infty^{\#}$, $\#\text{-lim}_{x\to x_0^+} f(x) = \infty^{\#}$.

**Example 11.1.** (i) $\#\text{-lim}_{x\to\#} x_0 - x^{-1} = -\infty^{\#}$, (ii) $\#\text{-lim}_{x\to\#} x_0 + x^{-1} = +\infty^{\#}$, (iii) $\#\text{-lim}_{x\to\#} -\infty^{\#} x^2 = \#\text{-lim}_{x\to\#} \infty^{\#} x^2 = \infty^{\#}$.

**Remark 11.1.** Throughout this paper, $\#\text{-lim}_{x\to\#} x_0 f(x)$ exists" will mean that $\#\text{-lim}_{x\to x_0} f(x) = L$, where $L$ is finite or hyperfinite.

To leave open the possibility that $L = \pm\infty^{\#}$, we will say that $\#\text{-lim}_{x\to x_0} f(x)$ exists in the extended hyperreals.

This convention also applies to one-sided limits and limits as $x$ approaches $\pm\infty^{\#}$.

## 11.2. Monotonic Functions $f : \mathbb{R}^{\#}_c \to \mathbb{R}^{\#}_c$.

**Definition 11.6.** A function $f : \mathbb{R}^{\#}_c \to \mathbb{R}^{\#}_c$ is nondecreasing on an interval $I \subset \mathbb{R}^{\#}_c$ if

$$
f(x_1) \leq f(x_2) \tag{11.6}
$$

whenever $x_1$ and $x_2$ are in $I$ and $x_1 < x_2$, or nonincreasing on $I$ if

$$
f(x_1) \geq f(x_2) \tag{11.7}
$$
whenever $x_1$ and $x_2$ are in I and $x_1 < x_2$.

In either case, $f$ is on I. If $\le$ can be replaced by $<$ in (11.6), $f$ is increasing on I. If $\ge$ can be replaced by $>$ in (11.7), $f$ is decreasing on I. In either of these two cases, $f$ is strictly monotonic on I.

**Theorem 11.4.** Suppose that $f(x)$ is monotonic on $(a, b)$ and define
$\alpha = \inf_{a<x<b} f(x)$ and $\beta = \sup_{a<x<b} f(x)$. Suppose that $\exists \alpha$ and $\exists \beta$, then:

(a) If $f$ is nondecreasing, then $f(a +) = \alpha$ and $f(b -) = \beta$.

(b) If $f$ is nonincreasing; then $f(a +) = \beta$ and $f(b -) = \alpha$.

Here $a += -\infty^{\#}$ if $a = -\infty^{\#}$ and $b += \infty^{\#}$ if $b = \infty^{\#}$.

(c) If $a < x_0 < b$, then $f(x_0 +)$ and $f(x_0 -)$ exist and are finite or hyperfinite;
moreover, $f(x_0 +) \le f(x_0) \le f(x_0 -)$ if $f$ is nondecreasing, and $f(x_0 +) \ge f(x_0) \ge f(x_0 -)$
if $f$ is nonincreasing:

**Proof** (a) We first show that $f(a +) = \alpha$. If $M > \alpha$, there is an $x_0$ in $(a, b)$ such that $f(x_0) < M$. Since $f$ is nondecreasing, $f(x) < M$ if $a < x < x_0$. Therefore, if $\alpha = -\infty^{\#}$, then $f(a +) = -\infty^{\#}$. If $\alpha > -\infty^{\#}$, let $M = \alpha + \varepsilon$, where $\varepsilon \approx 0, \varepsilon > 0$.

Then $\alpha \le f(x) < \alpha + \varepsilon$, so (i) $|f(x) - \alpha| < \varepsilon$ if $a < x < x_0$.

If $a = -\infty^{\#}$, this implies that $f(-\infty^{\#}) = \alpha$. If $a > -\infty^{\#}$, let $\delta = x_0 - a$. Then (i) is equivalent to $|f(x) - \alpha| < \varepsilon$ if $a < x < a + \delta$, which implies that $f(a +) = \alpha$.
We now show that $f(b +) = \beta$. If $M < \beta$, there is an $x_0$ in $(a, b)$ such that $f(x_0) > M$.

Since $f(x)$ is nondecreasing, $f(x) > M$ if $x_0 < x < b$. Therefore, if $\beta = \infty^{\#}$, then $f(b -) = \infty^{\#}$. If $\beta < \infty^{\#}$, let $M = \beta - \varepsilon$, where $\varepsilon \approx \varepsilon > 0$. Then $\beta - \varepsilon < f(x) \le \beta$, so (ii) $|f(x) - \beta| < \varepsilon$ if $x_0 < x < b$.

If $b = \infty^{\#}$, this implies that $f(\infty^{\#}) = \beta$. If $b < \infty^{\#}$, let $\delta = b - x_0$. Then (ii) is equivalent to $f(x) < b - \delta < x < b$, which implies that $f(b -) = \beta$.

(b) The proof is similar to the proof of (a).

(c) Suppose that $f(x)$ is nondecreasing. Applying (a) to $f(x)$ on $(a, x_0)$ and $(x_0, b)$
separately shows that $f(x_0 -) = \sup_{a<x<x_0} f(x)$ and $f(x_0 +) = \inf_{x_0<x<b} f(x)$.
However, if $x_1 < x_0 < x_2$, then $f(x_1) \le f(x_0) \le f(x_2)$ and hence, $f(x_0 -) \le f(x_0) \le f(x_0 +)$.

## 11.3. #-Limits Inferior and Superior

**Definition 11.7.** We say that: (i) $f$ is bounded on a set $S \subseteq \mathbb{R}_c^\#$ if there is a constant $M \in \mathbb{R}, M < \infty$ such that $f(x) \le M$ for all $x \in S$, (ii) $f$ is hyperbounded on a set $S \subseteq \mathbb{R}_c^\#$ if $f$ is not bounded on a set $S$ and there is a constant $M \in \mathbb{R}_c^\#/\mathbb{R}, M < \infty^\#$ such that $f(x) \le M$ for all $x \in S$.

**Definition 11.8.** Suppose that $f$ is bounded or hyperbounded on $[a, x_0)$, where $x_0$ may be finite or hyperfinite or $\infty^\#$. For $a \le x < x_0$, define (i) $S_f(x; x_0) = \sup_{x \le t < x_0} f(t)$ and (ii) $I_f(x; x_0) = \inf_{x \le t \le x_0} f(t)$.

Then the left #-limit superior of $f(x)$ at $x_0$ is defined to be

$$ \overline{\lim}_{x \to \#} x_0^- f(x) = \lim_{x \to \#} x_0^- S_f(x; x_0) \quad (11.8) $$

and the left limit inferior of $f(x)$ at $x_0$ is defined to be

$$ \underline{\lim}_{x \to \#} x_0^- f(x) = \lim_{x \to \#} x_0^- I_f(x; x_0). \quad (11.9) $$

If $x_0 = \infty^\#$, we define $x_0 = -\infty^\#$.

**Theorem 11.5.** If $f(x)$ is bounded or hyperbounded on $[a, x_0)$, then $\beta = \overline{\lim}_{x \to \#} x_0^- f(x)$ exists and is the unique hyperreal number with the following properties:
(a) If $\varepsilon > 0, \varepsilon \approx 0$, there is an $a_1$ in $[a, x_0)$ such that

(i) $f(x) < \beta + \varepsilon$ if $a_1 \le x < x_0$

(b) If $\varepsilon > 0, \varepsilon \approx 0$ and $a_1$ is in $[a, x_0)$, then

$f(\bar{x}) > \beta - \varepsilon$ for some $\bar{x} \in [a, x_0)$.

**Proof.** Since $f(x)$ is bounded or hyperbounded on $[a, x_0)$, $S_f(x; x_0)$ is nonincreasing and bounded or hyperbounded on $[a, x_0)$. By applying Theorem 11.4(b) to $S_f(x; x_0)$, we conclude that $\beta$ exists finite or hyperfinite.

Therefore, if $\varepsilon > 0, \varepsilon \approx 0$, there is an $\bar{a}$ in $[a, x_0)$ such that

(ii) $\beta - \varepsilon/2 < S_f(x; x_0) < \beta + \varepsilon/2$ if $\bar{a} \le x < x_0$.

Since $S_f(x; x_0)$ is an upper bound of $\{f(t) | x \le t < x_0\}$, $f(x) < S_f(x; x_0)$. Therefore, the second inequality in (ii) implies the inequality (i) with $a_1 = \bar{a}$. This proves (a).

To prove (b), let $a_1$ be given and define $x_1 = \max\{a_1, \bar{a}\}$. Then the first inequality in (ii) implies that (iii) $S_f(x; x_0) > \beta - \varepsilon/2$. Since $S_f(x; x_0)$ is the supremum of

$\{f(t) | x_1 \le t < x_0\}$, there is an $\bar{x}$ in $[x_1, x_0)$ such that

$f(\bar{x}) > S_f(x; x_0) - \varepsilon/2$. This and (iii) imply that $f(\bar{x}) > \beta - \varepsilon/2$. Since $\bar{x}$ is in $[a_1, x_0)$, this proves (b).

Now we show that there cannot be more than one hyperreal number with properties (a) and (b). Suppose that $\beta_1 < \beta_2$ and $\beta_2$ has property (b); thus, if $\varepsilon \approx 0, \varepsilon > 0$ and $a_1$ is in $[a, x_0)$ there is an $\bar{x}$ in $[a_1, x_0)$ such that $f(\bar{x}) > \beta_2 - \varepsilon$. Letting $\varepsilon = \beta_2 - \beta_1$, we see that there is an $\bar{x}$ in $[a_1, b)$ such that $f(\bar{x}) > \beta_2 - (\beta_2 - \beta_1) = \beta_1$ so $\beta_1$ cannot have property (a). Therefore, there cannot be more than one hyperreal number that satisfies both (a) and (b).

**Theorem 11.6.** If $f(x)$ is bounded or hyperbounded on $[a, x_0)$, then $\alpha = \lim_{x \to x_0^-} f(x)$ exists and there is the unique hyperreal number with the following properties:

(a) If $\varepsilon \approx 0, \varepsilon > 0$ there is an $a_1$ in $[a, x_0)$ such that

$f(x) > \alpha - \varepsilon$ if $a_1 \le x < x_0$.

(b) If $\varepsilon \approx 0, \varepsilon > 0$ and $a_1$ is in $[a, x_0)$, then

$f(\bar{x}) < \alpha + \varepsilon$ for some $\bar{x} \in [a, x_0)$.

**Theorem 11.7.** If $f(x)$ is bounded or hyperbounded on $[a, x_0)$, then

(i) $\underline{\lim}_{x \to x_0^-} f(x) \le \overline{\lim}_{x \to x_0^-} f(x)$;

(ii) $\underline{\lim}_{x \to x_0^-} f(-x) = -\overline{\lim}_{x \to x_0^-} f(x)$;

(iii) $\overline{\lim}_{x \to x_0^-} f(-x) = -\underline{\lim}_{x \to x_0^-} f(x)$;

(iv) $\underline{\lim}_{x \to x_0^-} f(x) = \overline{\lim}_{x \to x_0^-} f(x)$ if and only if $\lim_{x \to x_0^-} f(x)$ exists, in which case

$$\lim_{x \to x_0^-} f(x) = \underline{\lim}_{x \to x_0^-} f(x) = \overline{\lim}_{x \to x_0^-} f(x)$$

**Theorem 11.8.** Suppose that $f(x)$ and $g(x)$ are bounded or hyperbounded on $[a, x_0)$.

Then: (i) $\overline{\lim}_{x \to x_0^-} (f + g)(x) \le \overline{\lim}_{x \to x_0^-} f(x) + \overline{\lim}_{x \to x_0^-} g(x)$;

(ii) $\underline{\lim}_{x \to x_0^-} (f + g)(x) \ge \underline{\lim}_{x \to x_0^-} f(x) + \underline{\lim}_{x \to x_0^-} g(x)$.

**Theorem 11.9.** The $\alpha = \lim_{x \to x_0^-} f(x)$ exists i.e., $\alpha$ is finite or hyperfinite

if and only if for each $\varepsilon \approx 0, \varepsilon > 0$ there is a $\delta \approx 0, \delta > 0$

such that $|f(x_1) - f(x_2)| < \varepsilon$ if $x_0 - \delta < x_1, x_2 < x_0$.

**Theorem 11.10.** (i) Suppose that $f(x)$ is bounded or hyperbounded on an interval $(x_0, b]$,

then $\underline{\lim}_{x \to x_0^+} f(x) = \overline{\lim}_{x \to x_0^+} f(x)$ if and only if $\lim_{x \to x_0^+} f(x)$ exists, in which case

$$\lim_{x \to x_0^+} f(x) = \underline{\lim}_{x \to x_0^+} f(x) = \overline{\lim}_{x \to x_0^+} f(x).$$

(ii) Suppose that $f(x)$ is bounded or hyperbounded on an open interval containing $x_0$,
then $\lim_{x \to x_0} f(x)$ exists if and only if

$$ \overline{\lim}_{x \to x_0} f(x) = \underline{\lim}_{x \to x_0} f(x) = \underline{\lim}_{x \to x_0} f(x) = \underline{\lim}_{x \to x_0} f(x). $$

### 11.4. The #-continuity of a function $f: \mathbb{R}_c^\# \to \mathbb{R}_c^\#$.

**Definition 11.9.** (i) We say that a function $f: \mathbb{R}_c^\# \to \mathbb{R}_c^\#$. is #-continuous at $x_0$ if $f$ is defined on an open interval $(a, b)$ containing $x_0$ and $\lim_{x \to x_0} f(x) = x_0$.

(ii) We say that $f$ is #-continuous from the left at $x_0$ if $f$ is defined on an open interval $(a, x_0)$ and $f(x_0 -) = f(x_0)$.

(iii) We say that $f$ is #-continuous from the right at $x_0$ if $f$ is defined on an open interval $(x_0, b)$ and $f(x_0 +) = f(x_0)$.

**Theorem 11.11.** (i) A function $f$ is #-continuous at $x_0$ if and only if $f$ is defined on an open interval $(a, b)$ containing $x_0$ and for each $\varepsilon \approx 0, \varepsilon > 0$ there is a $\delta \approx 0, \delta > 0$ such that

$|f(x) - f(x_0)| < \varepsilon$

(11.10)

whenever $|x - x_0| < \delta$.

(ii) A function $f$ is #-continuous from the right at $x_0$ if and only if $f$ is defined on an interval $[x_0, b)$ and for each $\varepsilon \approx 0, \varepsilon > 0$ there is a $\delta \approx 0, \delta > 0$ such that (11.10) holds whenever $x_0 \le x < x_0 + \delta$.

(iii) A function $f$ is #-continuous from the left at $x_0$ if and only if $f$ is defined on an interval $(a, x_0]$ and for each $\varepsilon \approx 0, \varepsilon > 0$ there is a $\delta \approx 0, \delta > 0$ such that (11.10) holds whenever $x_0 - \delta < x \le x_0$.

Note that from Definition 11.9 and Theorem 11.8, $f$ is #-continuous at $x_0$ if and only if $f(x_0 +) = f(x_0 -) = f(x_0)$ or, equivalently, if and only if it is #-continuous from the right and left at $x_0$.

**Definition 11.10.** A function $f: \mathbb{R}_c^\# \to \mathbb{R}_c^\#$ is #-continuous on an open interval $(a, b)$ if it is

#-continuous at every point in $(a, b)$. If, in addition,

$$ f(b -) = f(b) \qquad (11.11) $$

or

$$ f(a +) = f(a) \qquad (11.12) $$

then $f$ is #-continuous on $(a, b]$ or $[a, b)$, respectively. If $f$ is #-continuous on $(a, b)$ and (11.11) and (11.12) both hold, then $f$ is #-continuous on $[a, b]$. More generally, if $S$ is a subset of $\text{dom}(f)$ consisting of finitely or countably or hyper finitely or hyper infinitely many disjoint intervals, then $f$ is #-continuous on $S$ if $f$ is #-continuous on every interval in $S$.

**Definition 11.11.** A function $f: \mathbb{R}_c^\# \to \mathbb{R}_c^\#$ is piecewise #-continuous on $[a, b]$ if

(i) $f(x_0 +)$ exists for all $x_0$ in $[a, b)$;

(ii) $f(x_0 -)$ exists for all $x_0$ in $(a, b]$;

(iii) $f(x_0 +) = f(x_0 -) = f(x_0)$ for all but except finitely or hyper finitely many points $x_0$ in $(a, b)$.

If (iii) fails to hold at some $x_0$ in $(a, b)$, $f$ has a jump #-discontinuity at $x_0$. Also, $f$ has a jump #-discontinuity at $a$ if $f(a +) \neq f(a)$ or at $b$ if $f(b -) \neq f(b)$.

**Theorem 11.12.** If $f$ and $g$ are #-continuous on a set $S$, then so are $f \pm g$, and $fg$. In addition, $fg$ is #-continuous at each $x_0$ in $S$ such that $g(x_0) \neq 0$.

By hyper infinite induction, it can be shown that if $\forall n \in \mathbb{N}^\# f_n(x)$ are #-continuous on a
set S, then so are $\sum_{i \le n} f_n(x)$. Therefore, $\forall n, m \in \mathbb{N}^\# \text{ any rational function}$

$r(x) = \sum_{i \le n} a_i x^i / \sum_{i \le m} b_i x^i, b_i \ne 0$ is \#-continuous for all values of $x$ except those for which its denominator vanishes.

## 11.5. Removable \#-discontinuities.

**Definition 11.12.** Let $f(x)$ be defined on a deleted \#-neighborhood of $x_0$ and \#-discontinuous (perhaps even undefined) at $x_0$. Then we say that $f(x)$ has a removable \#-discontinuity at $x_0$ if $\lim_{x \to x_0} f(x_0)$ exists. In this case, the function

$$ g(x) = \begin{cases} f(x) & \text{if } x \in \text{dom}(f) \text{ and } x \neq x_0 \\ \lim_{x \to x_0} f(x_0) & \text{if } x = x_0 \end{cases} \qquad (11.13) $$

is \#-continuous at $x_0$.

## 11.6. Composite Functions $f : \mathbb{R}_c^\# \to \mathbb{R}_c^\#$.

**Definition 11.13.** Suppose that $f : \mathbb{R}_c^\# \to \mathbb{R}_c^\#$ and $g : \mathbb{R}_c^\# \to \mathbb{R}_c^\#$ are functions with domains **dom**(f) and **dom**(g) correspondingly. If **dom**(g) has a nonempty subset T such that $g(x) \in \text{dom}(g)$ whenever $x \in T$, then the composite function $f \circ g : \mathbb{R}_c^\# \to \mathbb{R}_c^\#$ is defined

on T by $(f \circ g)(x) = f(g(x))$

**Theorem 11.10.** Suppose that $g$ is \#-continuous at $x_0$, $g(x_0)$ is an \#-interior point of **dom**(f) and f is \#-continuous at $g(x_0)$. Then $f \circ g$ is \#-continuous at $x_0$.

**Proof.** Suppose that $\varepsilon \approx 0, \varepsilon > 0$. Since $g(x_0)$ is an \#-interior point of **dom**(f) and $f(x)$ is \#-continuous at $g(x_0)$, there is a $\delta_1 \approx 0, \delta_1 > 0$ such that $f(t)$ is defined and

(i) $|f(t) - f(gx_0)| < \varepsilon$ if $|t - g(x_0)| < \delta_1$.

Since $g(x)$ is \#-continuous at $x_0$, there is a $\delta \approx 0, \delta > 0$ such that $g(x)$ is defined and

(ii) $|g(x) - g(x_0)| < \delta_1$ if $|x - x_0| < \delta$.

Now (i) and (ii) imply that $|f(g(x)) - f(g(x_0))| < \delta$ if $|x - x_0| < \varepsilon$. Therefore, $f \circ g$ is \#-continuous at $x_0$.

## 11.7. Bounded and Hyperbounded Functions $f : \mathbb{R}_c^\# \to \mathbb{R}_c^\#$.

**Definition 11.14.** (i) A function $f : \mathbb{R}_c^\# \to \mathbb{R}_c^\#$ is bounded below on a set $S \subset \mathbb{R}_c^\#$ if there is a finite or hyperfinite hyperreal number $m \in \mathbb{R}_c^\#$. Such that $f(x) \ge m$ for all $x \in S$. If in this case the set $V = \{f(x) | x \in S\}$ has infimum $\alpha$, we write $\alpha = \inf_{x \in S} f(x)$. If there is a point $x_1 \in S$ such that $f(x_1) = \alpha$, we say that $\alpha$ is the minimum of $f(x)$ on S, and write $\alpha = \min_{x \in S} f(x)$

(ii) A function $f : \mathbb{R}_c^\# \to \mathbb{R}_c^\#$ is bounded above on $S \subset \mathbb{R}_c^\#$ if there is a finite or hyperfinite hyperreal number $M \in \mathbb{R}_c^\#$. Such that $f(x) \le M$ for all $x \in S$. If in this case, V has a supremum $\beta$, we write $\beta = \sup_{x \in S} f(x)$. If there is a point $x_1 \in S$ such that $f(x_1) = \beta$, we say that $\beta$ is the maximum of $f(x)$ on S, and write $\alpha = \max_{x \in S} f(x)$.

(iii) If $f$ is bounded above and below on a set $S$, we say that $f$ is bounded on $S$.

**Theorem 11.11.** If $f$ is \#-continuous on a finite or hyperfinite \#-closed interval $[a, b]$,
then $f$ is bounded on $[a, b]$.

**Proof.** Suppose that $t \in [a, b]$. Since $f$ is #-continuous at $t$, there is an open interval $I_t$ containing $t$ such that

$$|f(x) - f(t)| < 1 \text{ if } x \in I_t \cap [a, b] \qquad (11.14)$$

To see this, set $\varepsilon = 1$ in (11.10), Theorem 11.11. The collection $H = \{I_t|a \le t \le b\}$ is an open covering of $[a, b]$. Since $[a, b]$ is #-compact, the generalized Heine-Borel theorem implies that there are hyper finitely many points $t_1, t_2, \dots, t_n, n \in \mathbb{N}^\#$ such that the intervals $I_{t_1}, I_{t_2}, \dots, I_{t_n}$ cover $[a, b]$. According to (11.14) with $t = t_i$, $|f(x) - f(t_i)| < 1$ if $x \in I_{t_i} \cap [a, b]$. Therefore,

$$f(x) = |(f(x) - f(t_i)) + f(t_i)| \le |f(x) - f(t_i)| + |f(t_i)| \le 1 + |f(t_i)| \qquad (11.15)$$

if $x \in I_{t_i} \cap [a, b]$. Let $M = 1 + \max_{1 \le j \le n}|f(t_i)|$. Since $[a, b] \subset \bigcup_{i=1}^n (I_{t_i} \cap [a, b])$, (11.15) implies that $|f(x)| \le M$ if $x \in [a, b]$.

**Theorem 11.12.** Suppose that $f$ is #-continuous on a finite or hyperfinite closed interval $[a, b]$. Let $V_{a,b} = \{f(x)|x \in [a, b]\}$ and let

$$\alpha = \inf V_{a,b} = \inf_{a \le x \le b} f(x) \text{ and } \beta = \sup V_{a,b} = \sup_{a \le x \le b} f(x). \qquad (11.16)$$

Then $\alpha$ and $\beta$ are respectively the minimum and maximum of $f$ on $[a, b]$; that is there are points $x_1$ and $x_2$ in $[a, b]$ such that $\alpha = f(x_1)$ and $\beta = f(x_2)$.

**Proof.** We show that $x_1$ exists. Note that the set $V_{a,b}$ is admissible below (above), since $f$ is #-continuous on $[a, b]$. Suppose that there is no $x_1$ in $[a, b]$ such that $f(x_1) = \alpha$. Then $f(x) > \alpha$ for all $x \in [a, b]$. We will show that this leads to a contradiction. Suppose that $t \in [a, b]$. Then $f(t) > \alpha$, so $f(t) > [f(t) + \alpha]/2 > \alpha$. Since $f$ is #-continuous at $t$, there is an open interval $I_t$ about $t$ such that

$$f(x) > \frac{f(t) + \alpha}{2} \qquad (11.17)$$

if $x \in I_t \cap [a, b]$. The collection $H = \{I_t|a \le t \le b\}$ is an open covering of $[a, b]$. Since $[a, b]$ is #-compact, the generalized Heine-Borel theorem implies that there are hyper finitely many points $t_1, t_2, \dots, t_n$ such that the intervals $I_{t_1}, I_{t_2}, \dots, I_{t_n}$ cover $[a, b]$.

Define $\alpha_1 = \min_{1 \le j \le n}[f(t_i) + \alpha]/2$. Then, since $[a, b] \subset \bigcup_{i=1}^n (I_{t_i} \cap [a, b])$, (11.17) implies that $f(t) > \alpha_1, a \le t \le b$. But $\alpha_1 > \alpha$, so this contradicts the definition of $\alpha$. Therefore, $f(x_1) = \alpha$ for some $x_1 \in [a, b]$.

## 11.8. Generalized Intermediate Value Theorem.

The next theorem shows that if $f$ is continuous on a finite closed interval $[a, b]$, then $f$ assumes every value between $f(a)$ and $f(b)$ as $x$ varies from $a$ to $b$.

**Theorem 11.13. (Generalized Intermediate Value Theorem)** Suppose that $f$ is #-continuous on $[a, b]$, $f(a) \ne f(b)$ and $f(a) < \mu < f(b)$. Then $f(c) = \mu$ for some $c \in (a, b)$.

**Proof.** Suppose that $f(a) < \mu < f(b)$. The set $S = \{x|(a \le x \le b) \land (f(x) \le \mu)\}$ is bounded and nonempty. Note that the set $S$ is admissible above, since $f$ is #-continuous on $[a, b]$ and therefore $\sup S$ exists. Let $c = \sup S$. We will show that $f(c) = \mu$. If $f(c) > \mu$, then $c > a$ and, since $f$ is #-continuous at $c$, there is an $\varepsilon > 0, \varepsilon \approx 0$ such that
$f(x) > \mu$ if $c - \varepsilon < x \leq c$. Therefore, $c$ is an upper bound for $S$, which contradicts the definition of $c$ as the supremum of $S$. If $f(c) < \mu$, then $c < b$ and there is an $\varepsilon > 0, \varepsilon \approx 0$ such that $f(x) < \mu$ for $c \leq x < c - \varepsilon$, so $c$ is not an upper bound for $S$. This is also a contradiction. Therefore, $f(c) = \mu$. The proof for the case where $f(b) < \mu < f(a)$ can be obtained by applying this result to $-f(x)$.

**Lemma.11.1.** If $f$ is #-continuous at $x_0$ and $f(x_0) > \mu$, then $f(x) > \mu$ for all $x$ in some #-neighborhood of $x_0$.

## 11.9. Uniform #-Continuity.

**Definition 11.15.** A function $f$ is uniformly #-continuous on a subset $S$ of its domain if, for every $\varepsilon > 0, \varepsilon \approx 0$ there is a $\delta > 0, \delta \approx 0$ such that $|f(x) - f(x')| < \varepsilon$ whenever $|x - x'| < \delta$ and $x, x' \in S$.

We emphasize that in this definition $\delta$ depends only on and $S$ and not on the particular choice of $x$ and $x'$, provided that they are both in $S$.

**Theorem 11.14.** If $f$ is #-continuous on a #-closed and bounded or hyperbounded interval $[a, b]$, then $f$ is uniformly #-continuous on $[a, b]$.

**Proof.** Suppose that $\varepsilon > 0, \varepsilon \approx 0$. Since $f$ is #-continuous on $[a, b]$, for each $t \in [a, b]$ there is a positive number $\delta_t$ such that

$$|f(x) - f(t)| < \varepsilon/2 \qquad (11.18)$$

if $|x - t| < \delta_t$ and $x \in [a, b]$. If $I_t = (t - \delta_t, t + \delta_t)$, the collection $H = \{I_t | t \in [a, b]\}$ is an open covering of $[a, b]$. Since $[a, b]$ is #-compact, the generalized Heine-Borel theorem implies that there are hyper finitely many points $t_1, t_2, \dots, t_n$ in $[a, b]$ such that $I_{t_1}, I_{t_2}, \dots, I_{t_n}$ cover $[a, b]$. Now define

$$\delta = \min\{\delta_{t_1}, \delta_{t_2}, \dots, \delta_{t_n}\}. \qquad (11.19)$$

We will show that if

$$|x - x'| < \delta \text{ and } x, x' \in [a, b] \qquad (11.20)$$

then $|f(x) - f(x')| < \varepsilon$. From the triangle inequality one obtains:

$$|f(x) - f(x')| = |(f(x) - f(t_r)) + (f(t_r) - f(x'))| \leq |f(x) - f(t_r)| + |f(t_r) - f(x')| \quad (11.21)$$

Since $I_{t_1}, I_{t_2}, \dots, I_{t_n}$ cover $[a, b]$, $x$ must be in one of these intervals. Suppose that $x \in I_{t_r}$ that is,

$$|x - t_r| \leq \delta_{t_r}. \qquad (11.22)$$

From (11.18) with $t = t_r$,

$$|f(x) - f(t_r)| \leq \frac{\varepsilon}{2}. \qquad (11.23)$$

From (11.20), (11.22), and the triangle inequality,

$$|x' - t_r| = |(x' - x) + (x - t_r)| \leq |x' - x| + |x - t_r| < \delta + \delta_{t_r} \leq 2\delta_{t_r}. \qquad (11.24)$$

Therefore, (11.18) with $t = t_r$ and $x$ replaced by $x'$ implies that

$$|f(x') - f(t_r)| \leq \frac{\varepsilon}{2}. \qquad (11.25)$$

Thus (11.25), (11.21) and (11.23) imply that $|f(x') - f(t_r)| \leq \varepsilon/2$.

## 11.10. Monotonic External Functions $f: \mathbb{R}_c^\# \to \mathbb{R}_c^\#$.

**Theorem 11.15.** If $f$ is monotonic and nonconstant on $[a, b]$, then $f$ is #-continuous
on $[a, b]$ if and only if its range **range** ($f$) = {$f(x)|x \in [a, b]$} is the \#-closed interval with endpoints $f(a)$ and $f(b)$.

**Theorem 11.16.** Suppose that $f$ is increasing and \#-continuous on $[a, b]$ and let $f(a) = c$ and $f(b) = d$. Then there is a unique function $g$ defined on $[c, d]$ such that

$$g(f(x)) = x, a \le x \le b, \qquad (11.26)$$

and

$$f(g(y)) = y, c \le y \le d. \qquad (11.27)$$

Moreover, $g$ is \#-continuous and increasing on $[c, d]$:

The function $g$ of Theorem 11.16 is the inverse of $f$, denoted by $f^{-1}$. Since (11.26) and (11.27) are symmetric in $f$ and $g$, we can also regard $f$ as the inverse of $g$, and denote it by $g^{-1}$.

### 11.11. The \#-derivative of a $\mathbb{R}^\#_c$-valued function $f: D \to \mathbb{R}^\#_c$.

A function $f: D \to \mathbb{R}^\#_c, D \subset \mathbb{R}^\#_c$ is differentiable at an interior point $x_0 \in D$ of its domain $D \subset \mathbb{R}^\#_c$ if the difference quotient

$$\frac{f(x) - f(x_0)}{x - x_0}, x \neq x_0 \qquad (11.28)$$

approaches a \#-limit as $x$ approaches $x_0$, in which case the \#-limit is called the \#-derivative

of $f$ at $x_0$, and is denoted by $f^{\#'}(x_0)$ or by $f^{\#}(x_0)$ or by $d^\#f(x_0)/d^\#x$ i.e.,

$$d^\#f(x_0)/d^\#x \triangleq f^{\#'}(x_0) = \#\lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} \qquad (11.29)$$

If $f$ is defined on an open set $S \subset \mathbb{R}^\#_c$, we say that $f$ is \#-differentiable on $S$ if $f$ is \#-differentiable at every point of $S$. If $f$ is \#-differentiable on $S$, then $f^{\#'}$ is a function on $S$.

We say that $f$ is \#-continuously \#-differentiable on $S$ if $f^{\#'}(x)$ is \#-continuous on $S$. If $f$ is \#-differentiable on a \#-neighborhood of $x_0$, it is reasonable to ask if $f^{\#'}(x)$ is \#-differentiable at $x_0$. If so, we denote the \#-derivative of $f^{\#'}$ at $x_0$ by $f^{\#'}(x_0)$. This is the

second \#-derivative of $f$ at $x_0$, and it is also denoted by $f^{(2)\#}(x_0)$. Continuing inductively, if $f^{(n-1)\#}$

is defined on a \#-neighborhood of $x_0$, then the $n$-th \#-derivative of $f$ at $x_0$, denoted by $f^{(n)\#}(x_0)$, where $n \in \mathbb{N}^\#$ or by $d^{n\#}f(x_0)/d^{n\#}x^n$ is the \#-derivative of $f^{(n-1)\#}(x)$ at $x_0$. For convenience we define the zeroth \#-derivative of $f$ to be $f$ itself; thus $f^{(0)\#} = f$.

**Example 11.1** If $n \in \mathbb{N}^\# \mathbb{N}$ is a positive hyperinteger and $f(x), x^n$ then

$$\frac{f(x) - f(x_0)}{x - x_0} = \frac{x^n - x_0^n}{x - x_0} = \frac{x - x_0}{x - x_0} \left( \operatorname{Ext} - \sum_{k=0}^{n-1} x^{n-k-1} \right). \qquad (11.30)$$

$$\text{Thus } f^{\#'}(x_0) = \#\lim_{x \to x_0} \operatorname{Ext} - \sum_{k=0}^{n-1} x^{n-k-1} = nx^{n-1}.$$

**Lemma 11.2.** If $f$ is \#-differentiable at $x_0$; then

$$f(x) = f(x_0) + [f^{\#'}(x_0) + E(x)](x - x_0), \qquad (11.31)$$

where $E(x)$ is defined on a \#-neighborhood of $x_0$ and $\#\lim_{x \to x_0} E(x) = E(x_0) = 0$.

**Proof.** Define
$$E(x) = \begin{cases} \frac{f(x) - f(x_0)}{x - x_0} - f'^{\#}(x_0) & x \in \mathbf{Dom}(f) \text{ and } x \neq x_0 \\ 0 & x = x_0 \end{cases} \quad (11.32)$$

Solving (11.32) for $f(x)$ yields (11.31) if $x \neq x_0$, and (11.31) is obvious if $x = x_0$.

Definition 11.29 implies that $\#\lim_{x \to x_0} E(x) = 0$. We defined $E(x_0) = 0$ to make $E(x)$ #-continuous at $x_0$. Since the right side of (11.32) is #-continuous at $x_0$, so is the left. This yields the following theorem.

**Theorem 11.17.** If $f$ is #-differentiable at $x_0$; then $f$ is #-continuous at $x_0$.

**Theorem 11.18.** If $f$ and $g$ are #-differentiable at $x_0$, then so are $f \pm g$ and $fg$ with

(a) $(f+g)'^{\#}(x_0) = f'^{\#}(x_0) + g'^{\#}(x_0)$;

(b) $(f-g)'^{\#}(x_0) = f'^{\#}(x_0) - g'^{\#}(x_0)$;

(c) $(fg)'^{\#}(x_0) = f^{\#}(x_0)g(x_0) + f(x_0)g^{\#}(x_0)$;

(d) The quotient $f/g$ is #-differentiable at $x_0$ if $g(x_0) \neq 0$ with

$$\left(\frac{f}{g}\right)'^{\#}(x_0) = \frac{f'^{\#}(x_0)g(x_0) - g'^{\#}(x_0)f(x_0)}{[g(x_0)]^2}.$$

(e) If $n \in \mathbb{N}^{\#}$ and $f_i$, $1 \le i \le n$ are #-differentiable at $x_0$, then so are $\text{Ext}-\sum_{i=1}^{n} f_i$ and

$$\left( \text{Ext}-\sum_{i=1}^{n} f_i(x_0) \right)'^{\#} = \text{Ext}-\sum_{i=1}^{n} f_i^{\#}(x_0).$$

(f) If $n \in \mathbb{N}^{\#}$ and $f^{(n)\#}(x_0)$, $g^{(n)\#}(x_0)$ exist, then so does $(f \times g)^{(n)\#}(x_0)$ and

$$(fg)^{(n)\#}(x_0) = \text{Ext}-\sum_{i=0}^{n} \binom{n}{i} f^{(i)\#}(x_0)g^{(n-i)\#}(x_0).$$

**Proof.** For the statements (a)-(d) the proof is straightforward. For the statements (e) and (f) immediately by hyper infinite induction.

**Theorem 11.19.** (The Chain Rule) Suppose that $g$ is #-differentiable at $x_0$ and $f$ is #-differentiable at $g(x_0)$. Then the composite function $h = f \circ g$ defined by $h(x) = f(g(x))$ is #-differentiable at $x_0$ with $h^{\prime\#}(x) = f^{\prime\#}(g(x_0))g^{\prime\#}(x_0)$.

**Definition 11.16.** If $f(x)$ is defined on $[x_0, b)$, the right-hand derivative of $f(x)$ at $x_0$ is defined to be

$$f_{+}^{\prime\#}(x_0) = \#\lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}, \quad (11.33)$$

if the #-limit exists, while if $f$ is defined on $(a, x_0]$, the left-hand derivative of $f(x)$ at $x_0$ is defined to be

$$f_{-}^{\prime\#}(x_0) = \#\lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}, \quad (11.34)$$

if the #-limit exists.

**Remark 11.2.** Note that $f(x)$ is #-differentiable at $x_0$ if and only if $f_{+}^{\prime\#}(x_0)$ and $f_{-}^{\prime\#}(x_0)$ exist and are equal, in which case $f_{+}^{\prime\#}(x_0) = f_{-}^{\prime\#}(x_0) = f_{+}^{\prime\#}(x_0)$.

**Definition 11.17.** We say that $f(x_0)$ is a local extreme value of $f(x)$ if there is a $\delta > 0$, $\delta \approx 0$ such that $f(x) - f(x_0)$ does not change sign on

$$(x_0 - \delta x_0 + \delta) \cap \text{dom}(f). \quad (11.35)$$

More specifically, $f(x_0)$ is a local maximum value of $f(x)$ if

$$f(x) \le f(x_0) \quad (11.36)$$
or a local minimum value of $f(x)$ if

$$f(x) \geq f(x_0) \qquad (11.37)$$

for all $x \in (x_0 - \delta x_0 + \delta) \cap \text{dom}(f)$. The point $x_0$ is called a local extreme point of $f(x)$, or, more specifically, a local maximum or local minimum point of $f(x)$.

**Theorem 11.20.** If $f(x)$ is #-differentiable at a local extreme point $x_0 \in \text{dom}(f)$ then $f'^#(x_0) = 0$.

**Theorem 11.21.** (Generalized Rolle's Theorem) Suppose that $f$ is #-continuous on the #-closed interval $[a, b]$ and #-differentiable on the #-open interval $(a, b)$ and $f(a) = f(b)$. Then $f'^#(c) = 0$ for some $c \in (a, b)$.

**Theorem 11.22.** (Intermediate Value Theorem for #-Derivatives) Suppose that $f(x)$ is #-differentiable on $[a, b]$, $f'^#(a) \neq f'^#(b)$ and $f'^#(a) < \mu < f'^#(b)$. Then $f'^#(c) = \mu$ for some $c \in (a, b)$.

**Theorem 11.23.** (Generalized Mean Value Theorem) If $f$ and $g$ are #-continuous on the #-closed interval $[a, b]$ and #-differentiable on the open interval $(a, b)$, then

$$[g(b) - g(a)]f'^#(c) = [f(b) - f(a)]g'^#(c) \qquad (11.38)$$

for some $c \in (a, b)$.

**Theorem 11.24.** (Mean Value Theorem) If $f$ is #-continuous on the #-closed interval $[a, b]$ and #-differentiable on the #-open interval $(a, b)$, then

$$f'^#(c) = \frac{f(b) - f(a)}{b - a} \qquad (11.39)$$

for some $c \in (a, b)$.

**Theorem 11.25.** If $f'^#(x)$ for all $x \in (a, b)$, then $f$ is constant on $(a, b)$.

**Theorem 11.26.** If $f'^#(x)$ exists for all $x \in (a, b)$ and does not change sign on $(a, b)$, then $f(x)$ is monotonic on $(a, b)$ increasing, nondecreasing, decreasing, or nonincreasing as: (i) $f'^#(x) > 0$, (ii) $f'^#(x) \geq 0$, (iii) $f'^#(x) < 0$, (iv) $f'^#(x) \leq 0$, respectively, for all $x \in (a, b)$.

**Theorem 11.27.** If $|f'^#(x)| < M, a < x < b$ then

$$|f(x) - f(x')| \leq M|x - x'|, \qquad (11.40)$$

where $x, x' \in (a, b)$.

**Definition 11.18.** A function that satisfies an inequality like (11.40) for all $x$ and $x'$ in an interval is said to satisfy a Lipschitz condition on the interval.

**Theorem 11.28.** (Generalized L'Hospital's Rule) Suppose that $f$ and $g$ are #-differentiable and $g'^#$ has no zeros on $(a, b)$. Let $\#\lim_{x \to \#} b-f(x) = \#\lim_{x \to \#} b-g(x)$ or $\#\lim_{x \to \#} b-f(x) = \pm\infty$ and $\#\lim_{x \to \#} b-g(x) = \pm\infty$ and suppose that

$$\# - \lim_{x \to \#} b - \frac{f'^#(x)}{g'^#(x)} = L, \qquad (11.41)$$

where $L \in \mathbb{R}^#_c$ or $L = \pm\infty$. Then

$$\# - \lim_{x \to \#} b - \frac{f(x)}{g(x)} = L, \qquad (11.42)$$

As we saw above in Lemma 11.2 if $f$ is #-differentiable at $x_0$; then
$$f(x) = f(x_0) + f^{\prime\#}(x_0)(x - x_0) + E(x)(x - x_0), \qquad (11.43)$$

where $\#\lim_{x \to x_0} E(x) = 0$. To generalize this result, we first restate it: the polynomial $P_1(x) = f(x_0) + f^{\prime\#}(x_0)(x - x_0)$ which is of degree $\le 1$ and satisfies $P_1(x_0) = f(x_0)$, $P_1^{\prime\#}(x) = f^{\prime\#}(x_0)$, approximates $f(x)$ so well near $x_0$ such that

$$\#\lim_{x \to x_0} \frac{f(x) - P_1(x)}{x - x_0} = 0. \qquad (11.44)$$

Now suppose that $f$ has $n$ $\#$-derivatives at $x_0$ and $P_n(x)$ is the polynomial of degree $n \in \mathbb{N}^{\#}\mathbb{N}$ such that

$$P_n^{(r)\#}(x_0) = f^{(r)\#}(x_0), \quad 0 \le r \le n. \qquad (11.45)$$

Since $P_n(x)$ is a polynomial of hyperfinite degree $n$, it can be written as

$$P_n(x) = \mathrm{Ext} - \sum_{i=0}^{n} a_i (x - x_0)^i \qquad (11.46)$$

where $a_0, \dots, a_n \in \mathbb{R}_c^\#$ are constants. Differentiating (11.46) gives $P_n^{(r)\#}(x_0) = r!a_r$, $0 \le r \le n$, so (11.45) determines $a_r$ uniquely as $a_r = f^{(r)\#}(x_0)/r!$, $0 \le r \le n$. Therefore,

$$P_n(x) = \mathrm{Ext} - \sum_{r=0}^{n} \frac{f^{(r)\#}(x_0)}{r!}. \qquad (11.47)$$

We call $P_n(x)$ the $n$-th Taylor hyper polynomial of $f(x)$ about $x_0$

**Theorem 11.29.** If $f^{(n)\#}(x_0)$ exists for some hyper integer $n \in \mathbb{N}^\#\mathbb{N}$ and $P_n(x)$ is the $n$-th Taylor hyper polynomial of $f$ about $x_0$, then

$$\#\lim_{x \to x_0} \frac{f(x) - P_n(x)}{(x - x_0)^n} = 0. \qquad (11.48)$$

**Theorem 11.30.** (Generalized Taylor's Theorem) Suppose that $f^{(n+1)\#}(x)$ exists on an $\#$-open interval $I$ about $x_0$, and let $x \in I$. Then the remainder $R_n(x) = f(x) - P_n(x)$ can be written as

$$R_n(x) = \frac{f^{(n+1)\#}(c)}{(n+1)!} (x - x_0)^n, \qquad (11.49)$$

where $c$ depends upon $x$ and is between $x$ and $x_0$.

## 11.12. The Riemann integral of a $\mathbb{R}_c^\#$-valued external function $f(x)$.

The Riemann integral is defined as $\#\text{-limit of Riemann hyperfinite sums of functions with respect to tagged partitions of an interval } [a, b] \subset \mathbb{R}_c^\#$ A tagged hyperfinite partition $P$ of a closed interval $[a, b]$ on the real line is a hyperfinite sequence

$$a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \dots \le x_{n-1} \le t_n \le x_n = b, \qquad (11.50)$$

where $n \in \mathbb{N}^\#\mathbb{N}$. This partitions the interval $[a, b]$ into $n$ sub-intervals $[x_{i-1}, x_i]$ indexed by $i \in \mathbb{N}^\#$, each of which is "tagged" with a distinguished point $t_i \in [x_{i-1}, x_i]$. Thus, any set of $n+1 \in \mathbb{N}^\#\mathbb{N}$ points satisfying (11.50) defines a partition $P$ of $[a, b]$, which we denote by $P = \{x_0, x_1, \dots, x_n\}$. A Riemann hyperfinite sum of a function $f$ with respect to such a tagged hyperfinite partition is defined as

$$I_n = \sum_{i=1}^{n} f(t_i) \Delta_i, \qquad (11.51)$$
where $n \in \mathbb{N}^*\mathbb{N}$. thus each term of the sum (11.51) is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the width of sub-interval, $\Delta_i = x_i - x_{i-1}$. The mesh($P$) of such a tagged partition is the width of the largest sub-interval formed by the partition, $\max_{i=1...n} \Delta_i$.

**Definition 11.19.** The Riemann integral of a function $f$ over the interval $[a, b]$ is equal to $I$ if for every $\varepsilon > 0$, $\varepsilon \approx 0$ there exists $\delta > 0$, $\delta \approx 0$ such that for any partition with distinguished points on $[a, b]$ whose mesh is less than $\delta$.

**Upper and Lower Integrals.**

**Definition 11.20.** $f$ is bounded (hyperbounded) on $[a, b]$ and $P = \{x_0, x_1, ..., x_n\}$ is a hyperfinite partition of $[a, b]$, let

$$M_j = \sup_{x_{j-1} \le x \le x_j} f(x) \qquad (11.52)$$

and

$$m_j = \inf_{x_{j-1} \le x \le x_j} f(x) \qquad (11.53)$$

The upper external hyperfinite sum of $f$ over $P$ is

$$S(P) = \mathrm{Ext} - \sum_{j=1}^{n} M_j(x_j - x_{j-1}) \qquad (11.54)$$

and the upper external integral of $f$ over $[a, b]$, denoted by

$$\mathrm{Ext} - \int_{a}^{b} f(x) d^{\#}x \qquad (11.55)$$

is the infimum of all hyperfinite upper sums. The lower external hyperfinite sum of $f$ over $P$ is

$$s(P) = \mathrm{Ext} - \sum_{j=1}^{n} m_j(x_j - x_{j-1}) \qquad (11.56)$$

and the lower external integral of $f$ over $[a, b]$, denoted by

$$\mathrm{Ext} - \int_{a}^{b} f(x) d^{\#}x \qquad (11.57)$$

is the supremum of all lower hyperfinite sums. If $m \le f(x) \le M$ for all $x \in [a, b]$, then

$$m(b-a) \le s(P) \le S(P) \le M(b-a) \qquad (11.58)$$

for every hyperfinite partition $P$; thus, the set of upper hyperfinite sums of $f$ over all partitions $P$ of $[a, b]$ is bounded, as is the set of lower hyperfinite sums. Therefore, Theorems 1.1.3 and 1.1.8 imply that: if the quantity (11.55) and (11.57) exist then both are unique, and satisfy the inequalities

$$m(b-a) \le \mathrm{Ext} - \int_{a}^{b} f(x) d^{\#}x \le M(b-a) \qquad (11.59)$$

and

$$m(b-a) \le \mathrm{Ext} - \int_{a}^{b} f(x) d^{\#}x \le M(b-a). \qquad (11.60)$$

**Theorem 11.31.** Let $f$ be bounded on $[a, b]$, and let $P$ be a hyperfinite partition of $[a, b]$. Then (i) The upper hyperfinite sum $S(P)$ of $f$ over $P$ is the supremum of the set of all hyperfinite Riemann sums of $f$ over $P$.

(ii) The lower hyperfinite sum $s(P)$ of $f$ over $P$ is the infimum of the set of all hyperfinite Riemann sums of $f$ over $P$.
**Proof (a)** If $P = \{x_0, x_1, \dots, x_n\}$, then $S(P) = \text{Ext}-\sum_{j=1}^n M_j(x_j - x_{j-1})$ where
$M_j = \sup_{x_{j-1} \le x \le x_j} f(x)$.

An arbitrary hyperfinite Riemann sum of $f$ over $P$ is of the following form

$$ \sigma = \text{Ext}-\sum_{j=1}^{n} f(c_j)(x_j - x_{j-1}), \qquad (11.61) $$

where $x_{j-1} \le c_j \le x_j$. Since $f(c_j) \le M_j$, it follows that $\sigma \le S(P)$. Now let $\varepsilon > 0$, $\varepsilon \approx 0$ and choose $\bar{c}_j \in [x_{j-1}, x_j]$ so that

$$ f(\bar{c}_j) > M_j - \frac{\varepsilon}{n(x_j - x_{j-1})}, \qquad (11.62) $$

where $1 \le j \le n \in \mathbb{N}^{\#\mathbb{N}}$. The hyperfinite Riemann sum $\bar{\sigma}$ produced in this way is

$$ \bar{\sigma} = \text{Ext}-\sum_{j=1}^{n} f(\bar{c}_j)(x_j - x_{j-1}) > \text{Ext}-\sum_{j=1}^{n} \left[ M_j - \frac{\varepsilon}{n(x_j - x_{j-1})} \right] (x_j - x_{j-1}) = S(P) - \varepsilon. \qquad (11.63) $$

Now Theorem 1.1.3 implies that $S(P)$ is the supremum of the set of hyperfinite Riemann sums of $f$ over $P$.

## The Riemann–Stieltjes Integral of a $\mathbb{R}^\#_c$-valued external function $f(x)$.

**Definition 11.21.** Let $f$ and $g$ be defined on $[a, b]$. We say that $f$ is Riemann–Stieltjes integrable with respect to $g$ on $[a, b]$, if there is a number $L \in \mathbb{R}^\#_c$ with the following property: For every $\varepsilon > 0$, $\delta \approx 0$, there is a $\delta > 0$, $\delta \approx 0$ such that

$$ \left| \text{Ext}-\sum_{j=1}^{n} f(c_j)[g(x_j) - g(x_{j-1})] - L \right| < \varepsilon \qquad (11.64) $$

provided only that $P = \{x_0, x_1, \dots, x_n\}$, $n \in \mathbb{N}^{\#\mathbb{N}}$ is a hyperfinite partition of $[a, b]$ such that $\|P\| < \delta$ and $x_{j-1} \le c_j \le x_j, j \in n$. In this case, we say that $L$ is the external Riemann–Stieltjes integral of $f$ with respect to $g$ over $[a, b]$, and write

$$ \text{Ext}-\int_{a}^{b} f(x)d^{\#}g(x) = L. \qquad (11.65) $$

## 11.13 Existence of the integral of a $\mathbb{R}^\#_c$-valued external function $f(x)$.

**Lemma 11.3** Suppose that

$$ |f(x)| \le M, a \le x \le b \qquad (11.66) $$

and let $P'$ be a hyperfinite partition of $[a, b]$ obtained by adding $r \in \mathbb{N}^{\#\mathbb{N}}$ points to a partition $P = \{x_0, x_1, \dots, x_n\}$, $n \in \mathbb{N}^{\#\mathbb{N}}$ of $[a, b]$. Then

$$ S(P) \ge S(P') \ge S(P) - 2Mr\|P\| \qquad (11.67) $$

and

$$ s(P) \le s(P') \le s(P) + 2Mr\|P\|. \qquad (11.68) $$

**Theorem 11.32.** If $f(x)$ is bounded on $[a, b]$, then