nyuuzyou/wikis huggingface dataset:
Equivalence of Definitions of Connected Topological Space/No Union of Separated Sets implies No Continuous Surjection to Discrete Two-Point Space

== Theorem ==
Let $T = \struct {S, \tau}$ be a topological space.
Let $T$ be such that there are no two non-empty separated sets whose union is $S$.
Then there exists no continuous surjection from $T$ onto a discrete two-point space.
== Proof ==
Let $T = \struct {S, \tau}$ be a topological space such that there are no two non-empty separated sets whose union is $S$.
Let $D = \struct {\set {0, 1}, \tau}$ be the discrete two-point space on $\set {0, 1}$.
 $f: T \to \set {0, 1}$ is a continuous surjection.
By definition of continuous mapping:
:$\map {f^{-1} } 0$ and $\map {f^{-1} } 1$ are open sets of $T$.
From the definition of a mapping:
:$\map {f^{-1} } 0 \cup \map {f^{-1} } 1 = S$
and
:$\map {f^{-1} } 0 \cap \map {f^{-1} } 1 = \O$
Then:
:$\map {f^{-1} } 0 = S \setminus \map {f^{-1} } 1$
and:
:$\map {f^{-1} } 1 = T \setminus \map {f^{-1} } 0$
are clopen.
From Closed Set equals its Closure they are their respective closures.
It follows from the definition that $\map {f^{-1} } 0$ and $\map {f^{-1} } 1$ are separated subsets of $T$ whose union is $S$.
Hence, , one of them must be empty, and the other one must be $S$.
Therefore $f$ is constant, and so is not a surjection.
This contradicts the original hypothesis.
That is, there exists no continuous surjection from $T$ onto a discrete two-point space.
Category:Equivalence of Definitions of Connected Topological Space
Equivalence of Definitions of Consistent Proof System

== Theorem ==
Let $\LL_0$ be the language of propositional logic.
Let $\mathscr P$ be a proof system for $\LL_0$.
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
=== Definition 1 implies Definition 2 ===
Suppose that $\neg \vdash_{\mathscr P} \phi$.
Suppose additionally that there is some logical formula $\psi$ such that:
:$\vdash_{\mathscr P} \psi, \neg \psi$
By the Rule of Explosion:
:$\psi, \neg \psi \vdash_{\mathscr P} \phi$
By Provable Consequence of Theorems is Theorem, we conclude:
:$\vdash_{\mathscr P} \phi$
in contradiction to our assumption.
=== Definition 2 implies Definition 1 ===
Suppose either $\phi$ or $\neg \phi$ is not a theorem of $\mathscr P$.
The implication follows trivially.
== Sources ==
* : $\S 4.4$: Conditions for an Axiom System: Theorem $3$, Theorem $4$
Category:Proof Systems
Category:Propositional Logic
Equivalence of Definitions of Consistent Set of Formulas

== Theorem ==
Let $\LL_0$ be the language of propositional logic.
Let $\mathscr P$ be a proof system for $\LL_0$.
Let $\FF$ be a collection of logical formulas.
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
=== Definition 1 implies Definition 2 ===
Suppose that $\FF \not \vdash_{\mathscr P} \phi$.
 that there exists $\psi$ such that $\FF \vdash_{\mathscr P} \psi$ and $\FF \vdash_{\mathscr P} \neg \psi$.
Then by the Rule of Explosion (Variant 3):
:$\psi, \neg \psi \vdash_{\mathscr P} \phi$
and therefore, combining the two above lines:
:$\FF \vdash_{\mathscr P} \phi$
which is a contradiction.
=== Definition 2 implies Definition 1 ===
According to definition 2, for every logical formula $\phi$ either $\phi$ or $\neg \phi$ is not a $\mathscr P$-provable consequence of $\FF$.
In particular, then, there exists a logical formula $\psi$ such that $\FF \not \vdash_{\mathscr P} \psi$.
== Sources ==
* : $\text{II}.11$ Some Strategies for Constructing Proofs: Lemma $\text{II}.11.3$
Category:Proof Systems
Category:Language of Propositional Logic
Equivalence of Definitions of Constant Polynomial

== Theorem ==
Let $R$ be a commutative ring with unity.
Let $P\in R[x]$ be a polynomial in one variable over $R$.
=== Definition 1 ===
=== Definition 2 ===
=== Definition 3 ===
== Proof ==
=== 1 iff 2 ===
This is by definition of coefficients and degree.
=== 1 iff 3 ===
Category:Constant Polynomials
Equivalence of Definitions of Continuity on Metric Spaces

== Theorem ==
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
=== Definition by Points implies Definition by Open Sets ===
Suppose that $f$ is continuous at every point $x \in A_1$.
Let $U \subseteq M_2$ be open in $M_2$.
Let $x \in f^{-1} \sqbrk U$.
Since $U$ is open in $M_2$:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} {\map f x; d_2} \subseteq U$
where $\map {B_\epsilon} {\map f x; d_2}$ denotes the open $\epsilon$-ball of $\map f x$ in $M_2$.
By the definition of continuity at a point:
:$\exists \delta \in \R_{>0}: f \sqbrk {\map {B_\delta} {x; d_1} } \subseteq \map {B_\epsilon} {\map f x; d_2}$
So:
:$f \sqbrk {\map {B_\delta} {x; d_1} } \subseteq U$
and so:
:$\map {B_\delta} {x; d_1} \subseteq f^{-1} \sqbrk U$
Thus $f^{-1} \sqbrk U$ is open in $M_1$.
=== Definition by Open Sets implies Definition by Points ===
Suppose $f$ is defined to be continuous in the sense that:
:for every $U \subseteq A_2$ which is open in $M_2$, $f^{-1} \sqbrk U$ is open in $M_1$.
Let $x \in A_1$.
Then by Open Ball of Point Inside Open Ball:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} {\map f x; d_2}$ is open in $M_2$
So by hypothesis, $f^{-1} \sqbrk {\map {B_\epsilon} {\map f x; d_2} }$ is open in $M_1$.
As $\map f x \in \map {B_\epsilon} {\map f x; d_2}$, it follows that:
:$x \in f^{-1} \sqbrk {\map {B_\epsilon} {\map f x; d_2} }$
So by hypothesis:
:$\exists \delta \in \R_{>0}: \map {B_\delta} {x; d_1} \subseteq f^{-1} \sqbrk {\map {B_\epsilon} {\map f x; d_2} }$
Then:
:$f \sqbrk {\map {B_\delta} {x; d_1} } \subseteq \map {B_\epsilon} {\map f x; d_2}$
Thus by the $\epsilon$-Ball definition, $f$ is continuous at $x$.
== Sources ==
* : Chapter $\text{III}$: Metric Spaces: More About Continuity
* : Chapter $2$: Metric Spaces: $\S 6$: Open Sets and Closed Sets: Theorem $6.3$
* : $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Proposition $2.3.13$
* : $4.8$
Category:Continuous Mappings on Metric Spaces
Equivalence of Definitions of Continuous Real Function at Point

== Theorem ==
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
=== $(1)$ implies $(2)$ ===
Let $f$ be a Continuous Real Function at Point by definition $1$.
Then by definition:
:$\ds \lim_{y \mathop \to x} \map f y = \map f x$
We have that:
:$\ds x = \lim_{y \mathop \to x} y$
and so it follows that:
:$\ds \map f x = \map f {\lim_{y \mathop \to x} y}$
That is:
:$\ds \lim_{y \mathop \to x} \map f y = \map f {\lim_{y \mathop \to x} y}$
Thus $f$ is a Continuous Real Function at Point by definition $2$.
=== $(2)$ implies $(1)$ ===
Let $f$ be a Continuous Real Function at Point by definition $2$.
Then by definition:
:$\ds \lim_{y \mathop \to x} \map f y = \map f {\lim_{y \mathop \to x} y}$
We have that:
:$\ds \lim_{y \mathop \to x} y = x$.
Thus:
:$\ds \map f {\lim_{y \mathop \to x} y} = \map f x$.
That is:
:$\ds \lim_{y \mathop \to x} \map f y = \map f x$
Thus $f$ is a Continuous Real Function at Point by definition $1$.
== Sources ==
* : $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.1$ Limits and Continuity
Category:Continuous Real Functions
Equivalence of Definitions of Contractible Space

== Theorem ==
=== Definition 1 ===
=== Definition 2 ===
== Lemma == 
When the category of topological spaces $\bf Top$ is endowed with the cartesian monoidal structure, the terminal object $*$ (that is, any singleton set, endowed with the discrete topology) acts as monoidal unit.
== Remark ==
By the lemma above, there is a bijection of sets $\hom(*,X)\cong X$ between the set of functions $* \to X$ and this bijection can be promoted to a homeomorphism transporting to $\hom(*,X)$ the topology on $X$.
Note that the Lemma above and the subsequent remark hold in full generality, even when we do not restrict to a cartesian closed subcategory of $\bf Top$.
== Proof of the equivalence ==
Assume that $X$ is contractible according to definition 1.
Then every $x_0 \in X$ is represented by a unique $\ceiling {x_0}: *\to X$ such that the composition $X \overset {t_X}\to * \overset {\ceiling {x_0} }\to X$ is the constant mapping $X \to X : x \mapsto x_0$.
By assumption, there exists an $x_0$ such that the constant mapping is homotopic to the identity mapping on $X$, which means that $t_X$ and $\ceiling {x_0}$ are mutually inverse homotopy equivalences (the composition $*\to X\to *$ is evidently the identity of $\set {x_0}$).
Similarly, assume that $X$ is contractible according to definition 2.
This means that the terminal map $t_X : X \to *$ has an homotopy inverse $p : *\to X$ such that $X\to *\to X$ is homotopic to the identity of $X$.
But according to the lemma above, there is a unique $x_0\in X$ such that $p = \ceiling {x_0}$.
So again the composition $X \overset {t_X} \to * \overset {\ceiling {x_0} }\to X$ is the constant mapping at $x_0$.
Equivalence of Definitions of Convergence in Normed Division Rings

== Theorem ==
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$ be a sequence in $R$.
=== Definition 1 ===
=== Definition 2 ===
=== Definition 3 ===
=== Definition 4 ===
== Proof ==
=== Definition 1 iff Definition 2 ===
By definition, the metric induced by the norm $\norm {\, \cdot \,}$ is the mapping $d: R \times R \to \R_{\ge 0}$ defined as:
:$\map d {x, y} = \norm {x - y}$
From Metric Induced by Norm on Normed Division Ring is Metric, $d$ is a metric.
By definition of a convergent sequence in a metric space, $\sequence{x_n}$ converges to $x \in R$ :
:$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \map d {x_n, x} < \epsilon$
:
:$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x} < \epsilon$
The result follows.
=== Definition 1 iff Definition 3 ===
Let $x \in R$.
By definition of norm on a division ring, $\norm {\, \cdot \,}$ is a mapping $\norm {\, \cdot \,}:R \to \R_{\ge 0}$.
Then:
:$\forall n \in \N: \size {\norm{x_n - x} - 0} = \size {\norm{x_n - x}} = \norm{x_n - x}$
By definition of convergence of a real sequence, the real sequence $\sequence {\norm {x_n - x} }$ converges to $0$ in the reals $\R$ 
:$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {\norm{x_n - x} - 0} < \epsilon$
:
:$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \norm{x_n - x} < \epsilon$
The result follows.
=== Definition 2 iff Definition 4 ===
By definition, the metric induced by the norm $\norm {\, \cdot \,}$ is the mapping $d: R \times R \to \R_{\ge 0}$ defined as:
:$\map d {x, y} = \norm {x - y}$
From Metric Induced by Norm on Normed Division Ring is Metric, $d$ is a metric.
By definition, the topology induced by the norm $\norm {\, \cdot \,}$ is the topology $\tau$ induced by the metric $d$.
From Metric Induces Topology, $\tau$ is a topology.
 
From Sequence Converges to Point Relative to Metric iff it Converges Relative to Induced Topology:
:$\sequence {x_n}$ converges to $x$ in the metric induced by the norm $\norm {\, \cdot \,}$ $\sequence {x_n}$ converges to $x$ in the topology induced by the norm $\norm {\, \cdot \,}$ 
== Sources ==
* : $\S 1.2$: Normed Fields, Definition $1.7$
Category:Sequences
Category:Convergence
Category:Normed Division Rings
Equivalence of Definitions of Convergent Complex Sequence

== Theorem ==
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
=== $(1)$ implies $(2)$ ===
Let $\sequence {z_n}$ be a convergent complex sequence by definition 1.
Then by definition:
:$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$
Let $z_n = x_n + i y_n$.
Let $c = a + i b$.
Let $\epsilon \in \R_{>0}$.
Let $N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$.
Then:
Thus $\sequence {z_n}$ is a convergent complex sequence by definition 2.
=== $(2)$ implies $(1)$ ===
Let $\sequence {z_n} = \sequence {x_n + y_n}$ be a convergent complex sequence by definition 2.
Then by definition:
:$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \size {x_n - a} < \epsilon \text { and } \size {y_n - b} < \epsilon$
where $a + i b = c$.
Let $\epsilon \in \R_{>0}$.
Let $N \in \R: n > N \implies \size {x_n - a} < \dfrac \epsilon 2 \text { and } \size {y_n - b} < \dfrac \epsilon 2$.
Then:
Thus $\sequence {z_n}$ is a convergent complex sequence by definition 1.
== Sources ==
* : $\S 4.2$. Sequences: Theorem.
Category:Convergent Complex Sequences
Equivalence of Definitions of Convergent P-adic Sequence

== Theorem ==
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\sequence {x_n} $ be a sequence in $\Q_p$.
=== Definition 1 ===
=== Definition 2 ===
=== Definition 3 ===
=== Definition 4 ===
== Proof ==
From P-adic Numbers form Non-Archimedean Valued Field:
:the $p$-adic norm is the norm on a division ring.
By definition, the $p$-adic metric is the metric induced by the $p$-adic norm.
From Equivalence of Definitions of Convergence in Normed Division Rings, it follows that 
Definition 2, Definition 3 and Definition 4 are equivalent.
By definition of convergence in a normed division ring, $\sequence {x_n}$ converges to $x$ in the $p$-adic norm :
:$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x}_p < \epsilon$
So Definition 1 is equivalent to Definition 2.
Category:Sequences
Category:Convergence
Category:P-adic Number Theory
Equivalence of Definitions of Convergent Sequence in Metric Space

== Theorem ==
Let $M = \struct {A, d}$ be a metric space or a pseudometric space.
Let $\sequence {x_k}$ be a sequence in $A$.
=== Definition 1 ===
=== Definition 2 ===
=== Definition 3 ===
=== Definition 4 ===
== Proof ==
=== Definition 1 iff Definition 2 ===
=== Definition 1 iff Definition 3 ===
=== Definition 2 implies Definition 4 ===
=== Definition 4 implies Definition 2 ===
Category:Metric Spaces
Category:Convergence
Category:Sequences
Category:Convergent Sequences (Metric Space)
Category:Equivalence of Definitions of Convergent Sequence in Metric Space
Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 2

== Theorem ==
Let $M = \struct {A, d}$ be a metric space or a pseudometric space.
Let $\sequence {x_k}$ be a sequence in $A$.
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
By definition of an open ball:
:$\forall n \in \N: \map d {x_n, l} < \epsilon \iff x_n \in \map {B_\epsilon} l$
The result follows.
Category:Equivalence of Definitions of Convergent Sequence in Metric Space
Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 3

== Theorem ==
Let $M = \struct {A, d}$ be a metric space or a pseudometric space.
Let $\sequence {x_k}$ be a sequence in $A$.
=== Definition 1 ===
=== Definition 3 ===
== Proof ==
By definition of a convergent real sequence:
:$\ds \lim_{n \mathop \to \infty} \map d {x_n, l} = 0$
:$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {\map d {x_n, l} - 0} < \epsilon$ 
From Distance in Pseudometric is Non-Negative:
:$\forall x, y \in A: \map d {x, y} \ge 0$
Hence:
:$\forall n \in \N: \map d {x_n, l} = \size {\map d {x_n, l}} = \size {\map d {x_n, l} - 0}$
The result follows.
Category:Equivalence of Definitions of Convergent Sequence in Metric Space
Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 2 implies Definition 4

== Theorem ==
Let $M = \struct {A, d}$ be a metric space or a pseudometric space.
Let $l \in A$.
Let $\sequence {x_k}$ be a sequence in $A$.
Let $\sequence {x_k}$ satisfy:
:$\forall \epsilon > 0: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$
where $\map {B_\epsilon} l$ is the open $\epsilon$-ball of $l$.
Then:
:for every $\epsilon \in \R{>0}$, the open $\epsilon$-ball about $l$ contains all but finitely many of the $p_n$.
== Proof ==
Let a fixed $\epsilon \in \R{>0}$ be selected.
Then:
:$\exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$
Hence the only $x_k$ that cannot be in the open $\epsilon$-ball $\map {B_\epsilon} l$ of $l$ are those for which $n \le N$.
There are finitely many of these.
== Sources ==
* : $\S 3$ Numerical Sequences and Series, Theorem $3.2a$
Category:Equivalence of Definitions of Convergent Sequence in Metric Space
Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 4 implies Definition 2

== Theorem ==
Let $M = \struct {A, d}$ be a metric space or a pseudometric space.
Let $l \in A$.
Let $\sequence {x_k}$ be a sequence in $A$.
Let $\sequence {x_k}$ satisfy:
:for every $\epsilon \in \R{>0}$, the open $\epsilon$-ball about $l$ contains all but finitely many of the $p_n$.
Then:
:$\forall \epsilon > 0: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$
where $\map {B_\epsilon} l$ is the open $\epsilon$-ball of $l$.
== Proof ==
Let $\map {B_\epsilon} l$ be any open $\epsilon$-ball of $l$.
Let $A = \set {n: x_n \notin \map {B_\epsilon} l}$.
By assumption $A$ is finite.
From Finite Non-Empty Subset of Totally Ordered Set has Smallest and Greatest Elements, any finite subset of $\N$ has a maximum.
Let $N$ be the maximum of $A$.
Then for every $n > N$, $x_n$ must be in the open $\epsilon$-ball $\map {B_\epsilon} l$.
== Sources ==
* : $\S 3$ Numerical Sequences and Series, Theorem $3.2a$
Category:Equivalence of Definitions of Convergent Sequence in Metric Space
Equivalence of Definitions of Convergent Sequence in Topology

== Theorem ==
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
=== $(1)$ implies $(2)$ ===
Suppose that $\alpha \in S$ has:
:$\forall U \in \tau: \alpha \in U \implies \paren {\exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in U}$
Fix $U \in \tau$ with $\alpha \in U$. 
Then there exists $N \in \N$ such that $x_n \in U$ for $n > N$. 
So $x_n \not \in U$ for $n \in \N$ implies that $n \le N$.
That is: 
:$\set {n \in \N : x_n \not \in U} \subseteq \set {n \in \N : n \le N}$ 
Since $\set {n \in \N : n \le N}$ is finite, it follows that $\set {n \in \N : x_n \not \in U}$ is finite from Subset of Finite Set is Finite.
=== $(2)$ implies $(1)$ ===
Suppose that $\alpha \in S$ is such that: 
:$\forall U \in \tau: \alpha \in U \implies \set {n \in \N: x_n \notin U}$ is finite.
Fix $U \in \tau$ with $\alpha \in U$. 
Then: 
:$\set {n \in \N : x_n \notin U}$ is finite.
From Subset of Naturals is Finite iff Bounded, there exists $N \in \N$ such that: 
:$\set {n \in \N : x_n \notin U} \subseteq \set {n \in \N : n \le N}$
So, if $n > N$, we must have $x_n \in U$. 
Category:Convergent Sequences
Category:Topology
Equivalence of Definitions of Convex Hull

== Theorem ==
Let $V$ be a vector space over $\R$. 
Let $U \subseteq V$. 
=== Definition $1$ ===
=== Definition $2$ ===
=== Definition $3$ ===
== Proof ==
=== Definition $(1)$ iff Definition $(3)$ ===
This is demonstrated in Convex Hull is Smallest Convex Set containing Set.
=== Definition $(2)$ iff Definition $(3)$ ===
Let $\bigcap C$ denote the intersection of all convex subsets of $V$ containing $U$.
From Intersection of Convex Sets is Convex Set (Vector Spaces), $\bigcap C$ is a convex subset of $V$.
Let $S$ be the smallest convex subset of $V$ containing $U$
By Intersection is Largest Subset:
:$\bigcap C = S$
That is, a convex hull by definition $2$ is the same thing as a convex hull by definition $3$.
== Sources ==
* 
Category:Convex Hulls
Equivalence of Definitions of Convex Polyhedron

== Theorem ==
=== Definition 1 ===
=== Definition 2 ===
=== Definition 3 ===
=== Definition 4 ===
== Proof ==
=== $(1)$ implies $(2)$ ===
Let $P$ be a convex polyhedron by definition $1$.
Thus $P$ is a convex polyhedron by definition $2$.
=== $(2)$ implies $(1)$ ===
Let $P$ be a convex polyhedron by definition $2$.
Thus $P$ is a convex polyhedron by definition $1$.
Category:Convex Polyhedra
Equivalence of Definitions of Convex Real Function

== Theorem ==
Let $f$ be a real function which is defined on a real interval $I$.
=== Definition 1 ===
=== Definition 2 ===
=== Definition 3 ===
== Proof ==
Let $f$ be convex real function on $I$ according to definition 1.
That is:
:$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \le \alpha \map f x + \beta \map f y$
, assume $x \le y$.
Make the substitutions $x_1 = x, x_2 = \alpha x + \beta y, x_3 = y$.
As $\alpha + \beta = 1$, we have $x_2 = \alpha x_1 + \paren {1 - \alpha} x_3$.
Thus:
:$\alpha = \dfrac {x_3 - x_2} {x_3 - x_1}, \beta = \dfrac {x_2 - x_1} {x_3 - x_1}$
So:
Again from $(1)$:
Thus:
:$\dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \le \dfrac {\map f {x_3} - \map f {x_1} } {x_3 - x_1} \le \dfrac {\map f {x_3} - \map f {x_2} } {x_3 - x_2}$
So:
:$\dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \le \dfrac {\map f {x_3} - \map f {x_1} } {x_3 - x_1}$
demonstrating that $f$ is convex on $I$ according to definition 3, and:
:$\dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \le \dfrac {\map f {x_3} - \map f {x_2} } {x_3 - x_2}$
demonstrating that $f$ is convex on $I$ according to definition 2.
As each step is an equivalence, the argument reverses throughout.
== Sources ==
* : $\S 12.21 \ (2)$
Category:Convex Real Functions
Equivalence of Definitions of Convex Set (Order Theory)

== Theorem ==
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
=== $(1)$ implies $(2)$ ===
Let $A \subseteq S$ be convex in $S$ by definition 1.
Then by definition:
:$\forall x, y \in A: \forall z \in S: x \preceq z \preceq y \implies z \in A$
Let $x, y, z \in A$.
Let $x \prec z \prec y$.
Then by definition:
:$x \preceq z \preceq y$
and so by hypothesis $z \in A$.
That is:
:$\forall x, y \in A: \forall z \in S: x \prec z \prec y \implies z \in A$
Thus $A \subseteq S$ is convex in $S$ by definition 2.
=== $(2)$ implies $(1)$ ===
Let $A \subseteq S$ be convex in $S$ by definition 2.
Then by definition:
:$\forall x, y \in A: \forall z \in S: x \prec z \prec y \implies z \in A$
Let $x, y, z \in A$.
Let $x \preceq z \preceq y$.
First suppose $z \ne x$ and $z \ne y$.
Then by definition of the relation $\prec$:
:$x \prec z \prec y$
and so by hypothesis $z \in A$.
Let $x = z$ or $x = y$
We have that $x \in A$ and $y \in A$.
So $z \in A$.
That is, whether $x \prec z \prec y$ or $z = x$ or $x = y$:
:$\forall x, y \in A: \forall z \in S: x \preceq z \preceq y \implies z \in A$
Thus $A \subseteq S$ is convex in $S$ by definition 1.
Category:Convex Sets (Order Theory)
Equivalence of Definitions of Convex Set in Vector Space

== Theorem ==
Let $\Bbb F \in \set {\R, \C}$.
Let $V$ be a vector space over $\Bbb F$.
Let $C \subseteq V$.
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
From the definition of the linear combination $t C + \paren {1 - t} C$, we have: 
:$t C + \paren {1 - t} C = \set {t x + \paren {1 - t} y : x, y \in C}$
for $t \in \R$. 
So we have: 
:$t C + \paren {1 - t} C \subseteq C$
:
:$t x + \paren {1 - t} y \in C$ for all $x, y \in C$ and $t \in \closedint 0 1$.
Hence the result.
Category:Convex Sets (Vector Spaces)
Equivalence of Definitions of Coreflexive Relation

== Theorem ==
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
=== Definition 1 implies Definition 2 ===
Let $\RR$ be a relation which fulfils the condition:
:$\forall x, y \in S: \tuple {x, y} \in \RR \implies x = y$
Then:
Hence $\RR$ is coreflexive by definition 2.
=== Definition 2 implies Definition 1 ===
Let $\RR$ be a relation which fulfils the condition:
:$\RR \subseteq \Delta_S$
Then:
Hence $\RR$ is coreflexive by definition 1.
Category:Coreflexive Relations