nyuuzyou/wikis huggingface dataset:
Characterisation of Cauchy Sequence in Non-Archimedean Norm/Corollary 1

== Theorem ==
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.
Let $\sequence {x_n}$ be a sequence of integers such that:
:$\forall n: x_{n + 1} \equiv x_n \pmod {p^n}$
Then:
:$\sequence {x_n}$ is a Cauchy sequence in $\struct {\Q, \norm {\,\cdot\,}_p}$.
== Proof ==
By hypothesis:
:$\forall n \in \N: p^n \divides \paren {x_{n + 1} - x_n}$
By the definition of the $p$-adic norm:
:$\forall n \in \N: \norm {x_{n + 1} - x_n}_p \le \dfrac 1 {p^n}$
From Sequence of Powers of Number less than One:
:$\ds \lim_{n \mathop \to \infty} \dfrac 1 {p^n} = 0$
From the Squeeze Theorem for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \norm{x_{n+1} - x_n}_p = 0$.
From P-adic Norm on Rational Numbers is Non-Archimedean Norm, the $p$-adic norm is non-Archimedean.
From Characterisation of Cauchy Sequence in Non-Archimedean Norm:
:$\sequence {x_n}$ is a Cauchy sequence in $\struct {\Q, \norm {\,\cdot\,}_p}$.
Category:Characterisation of Cauchy Sequence in Non-Archimedean Norm
Characterisation of Cauchy Sequence in Non-Archimedean Norm/Necessary Condition

== Theorem ==
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a Cauchy sequence in $R$.
Then:
:$\lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$
== Proof ==
Let $\epsilon > 0$ be given.
By the definition of a Cauchy sequence:
:$\exists N: \forall n, m > N: \norm {x_n - x_m} < \epsilon$
So
:$\exists N: \forall n > N: \norm {x_{n + 1} - x_n} < \epsilon$
Hence the result follows:
:$\lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$.
== Sources ==
* : $\S 3.2$: Completions, Lemma $3.2.2$
Category:Characterisation of Cauchy Sequence in Non-Archimedean Norm
Characterisation of Cauchy Sequence in Non-Archimedean Norm/Sufficient Condition

== Theorem ==
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\ds \lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$.
Then:
:$\sequence {x_n}$ is a Cauchy sequence.
== Proof ==
Let $\epsilon > 0$ be given.
By assumption $\exists N \in \N$ such that:
:$(1) \quad \forall n > N: \norm {x_{n + 1} - x_n} < 0$
Suppose $n, m > N$, and $m = n + r > n$.
Then:
It follows that:
::$\sequence {x_n}$ is a Cauchy sequence.
== Sources ==
* : $\S 3.2$: Completions, Lemma $3.2.2$
Category:Characterisation of Cauchy Sequence in Non-Archimedean Norm
Characterisation of Jacobson Radical

== Theorem ==
Let $A$ be a commutative ring with unity.
Let $A^\times$ be the group of units of $A$.
Let $\map {\operatorname {Jac} } A$ be the Jacobson radical of $A$.
Then:
:$\map {\operatorname {Jac} } A = \set {a \in A: 1_A - a x \in A^\times \text{ for all } x \in A}$
where $1_A$ is the unity of $A$.
== Proof ==
Recall the definition of Jacobson radical of $A$:
:$\map {\operatorname {Jac} } A$ is the intersection of all maximal ideals of $R$.
=== $\subset$ Direction ===
 that $x \in \map {\operatorname {Jac} } A$ such that there exists $y \in A$ such that $1_A - x y \notin A^\times$. 
By definition $x$ is contained in all maximal ideals of $A$.
Let $\mathfrak m \subseteq A$ be one particular such maximal ideal of $A$.
Then $x \in \map {\operatorname {Jac} } A \subseteq \mathfrak m$ implies that $xy \in \mathfrak m$ and therefore $1_A \in \mathfrak m$.
But if $1_A \in \mathfrak m$ then from Ideal of Unit is Whole Ring: Corollary:
:$\mathfrak m = R$
This contradicts the definition of $\mathfrak m$ as a maximal ideal of $A$.
Hence by Proof by Contradiction:
:$\map {\operatorname {Jac} } A \subseteq \set {a \in A: 1_A - a x \in A^\times \text{ for all } x \in A}$
=== $\supset$ Direction ===
Now suppose that $x \notin \map {\operatorname {Jac} } A$.
Then $x \notin \mathfrak m$ for some maximal ideal $\mathfrak m$ of $A$.
Because $\mathfrak m$ is maximal, $x$ and $\mathfrak m$ generate $A$.
Therefore there exist $w \in \mathfrak m$ and $y \in A$ such that $w + x y = 1_A$.
Thus $1_A - x y \in \mathfrak m$, and $1_A - x y \notin A^\times$.
Hence:
:$\map {\operatorname {Jac} } A \supseteq \set {a \in A: 1_A - a x \in A^\times \text{ for all } x \in A}$
The result follows by definition of set equality.
Category:Commutative Algebra
Characterisation of Limit Element under Well-Ordering

== Theorem ==
Let $A$ be a class.
Let $\preccurlyeq$ be a well-ordering on $A$.
Let $x \in A$ be an element of $A$ such that $x$ is not the smallest element of $A$ under $\preccurlyeq$.
Then:
:$x$ is a limit element of $A$ under $\preccurlyeq$
:
:$x^\prec$ has no greatest element $\preccurlyeq$
where $x^\prec$ denotes the strict lower closure of $x$ in $A$ under $\preccurlyeq$.
== Proof ==
Suppose $x^\prec$ has a greatest element $y$.
Then $x$ is the immediate successor of $y$, that is:
:$\nexists z \in A : y < z < x$
Hence $x$ is not a limit element.
Therefore, if $x$ is a limit element, then $x^\prec$ cannot have a greatest element.
Suppose $x$ is an immediate successor.
Then the immediate predecessor of $x$ is the greatest element of $x^\prec$.
Therefore if $x^\prec$ has no greatest element then $x$ cannot be an immediate successor.
Hence if $x$ is not the smallest element of $A$, it must be a limit element.
== Sources ==
* : Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering: Proposition $1.6$
Category:Well-Orderings
Category:Limit Elements
Characterisation of Linearly Independent Set through Free Module Indexed by Set

== Theorem ==
Let $M$ be a unitary $R$-module.
Let $S = \family {m_i}_{i \mathop \in I}$ be a family of elements of $M$.
Let $\Psi : R^{\paren I} \to M$ be the module homomorphism given by Universal Property of Free Module on Set.
Then the following are equivalent:
:$S$ linearly independent
:$\Psi$ is injective.
== Proof ==
We have:
:$\map \Psi {\family {r_i}_{i \mathop \in I} } = 0$
:
:$\ds \sum_{i \mathop \in I} r_i m_i = 0$
Thus injectivity and linearly independent are equivalent.
Category:Module Theory
Characterisation of Local Rings

== Theorem ==
Let $R$ be a ring.
Let $J \lhd R$ be a maximal ideal.
:$(1): \quad$ If the set $R \setminus J$ is precisely the group of units $R^\times$ of $R$, then $\tuple {R, J}$ is a local ring.
:$(2): \quad$ If $1 + x$ is a unit in $R$ for all $x \in J$ then $\tuple {R, J}$ is local.
== Proof ==
$(1): \quad$ Suppose that $J$ is the set of non-units of $R$.
Then by Ideal of Unit is Whole Ring, every ideal not equal to $R$ is contained in $J$.
Therefore $J$ is the unique maximal ideal of $R$, so $\tuple {R, J}$ is local.
$(2): \quad$ Let $x \in R \setminus J$.
Then:
:$J \subsetneq \map I {J \cup \set x} \subseteq R$
where $\map I S$ is the ideal generated by $S$.
Since $J$ is maximal, by definition, $x$ and $J$ generate all of $R$.
Therefore $t x + m = 1$ for some $m \in J$, $t \in R$.
Thus $t x = 1 - m \in 1 + J$ is a unit by hypothesis.
Therefore $x$ is a unit.
Now use part $(1)$.
Category:Local Rings
Characterisation of Non-Archimedean Division Ring Norms

== Theorem ==
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.
Then $\norm {\,\cdot\,}$ is non-Archimedean :
:$\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$
where:
:$n \cdot 1_R = \underbrace {1_R + 1_R + \dotsb + 1_R}_{\text {$n$ times} }$
=== Corollary 1 ===
=== Corollary 2 ===
=== Corollary 3 ===
=== Corollary 4 ===
=== Corollary 5 ===
== Proof ==
=== Necessary Condition ===
=== Sufficient Condition ===
== Sources ==
* : $\S 2.2$ Basic Properties: Theorem $2.2.2$
* : $\S 1.2$ Normed Fields: Proposition $1.14$
Category:Normed Division Rings
Category:Characterisation of Non-Archimedean Division Ring Norms
Characterisation of Non-Archimedean Division Ring Norms/Corollary 1

== Theorem ==
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.
$\norm {\,\cdot\,}$ is non-Archimedean :
:$\sup \set {\norm {n \cdot 1_R}: n \in \N_{> 0}} = 1$.
where $n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} }$
== Proof ==
By Characterisation of Non-Archimedean Division Ring Norms then:
:$\norm {\,\cdot\,}$ is non-Archimedean :
::$\sup \set {\norm {n \cdot 1_R}: n \in \N_{\gt 0}} \le 1$
By norm of unity then:
:$\norm {1_R} = 1$
The result follows.
== Sources ==
* : $\S 2.2$ Basic Properties: Corollary $2.2.3$
Category:Characterisation of Non-Archimedean Division Ring Norms
Characterisation of Non-Archimedean Division Ring Norms/Corollary 2

== Theorem ==
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.
Let $\sup \set {\norm {n \cdot 1_R}: n \in \N_{> 0} } = C < +\infty$.
where $n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} }$
Then $\norm {\,\cdot\,}$ is non-Archimedean and $C = 1$.
== Proof ==
 $C > 1$.
By Characterizing Property of Supremum of Subset of Real Numbers:
:$\exists m \in \N_{> 0}: \norm {m \cdot 1_R} > 1$
Let 
:$x = m \cdot 1_R$ 
:$y = x^{-1}$
By Norm of Inverse:
:$\norm y < 1$
By Sequence of Powers of Number less than One:
:$\ds \lim_{n \mathop \to \infty} \norm y^n = 0$
By Reciprocal of Null Sequence then:
:$\ds \lim_{n \mathop \to \infty} \frac 1 {\norm y^n} = +\infty$
For all $n \in \N_{> 0}$:
So:
:$\ds \lim_{n \mathop \to \infty} \norm {m^n \cdot 1_R} = +\infty$
Hence:
:$\sup \set {\norm{n \cdot 1_R}: n \in \N_{> 0} } = +\infty$
This contradicts the assumption that $C < +\infty$.
It follows that $C \le 1$.
Then:
:$\forall n \in \N_{>0}: \norm{n \cdot 1_R} \le 1$
By Characterisation of Non-Archimedean Division Ring Norms, $\norm{\,\cdot\,}$ is non-Archimedean.
By Corollary 1:
:$\sup \set {\norm {n \cdot 1_R}: n \in \N_{> 0} } = 1$
So $C = 1$.
== Sources ==
* : $\S 2.2$ Basic Properties: Problem $41$
Category:Characterisation of Non-Archimedean Division Ring Norms
Characterisation of Non-Archimedean Division Ring Norms/Corollary 3

== Theorem ==
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with unity $1_R$.
$\norm {\,\cdot\,}$ is Archimedean :
:$\sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } = +\infty$
where:
:$n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} }$
== Proof ==
By Characterisation of Non-Archimedean Division Ring Norms:
:$\norm {\,\cdot\,}$ is Archimedean $\iff \sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } > 1$
By Corollary 2:
:$\sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } > 1 \iff \sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } = +\infty$
== Sources ==
* : $\S 2.2$ Basic Properties: Theorem $2.2.2$
* : $\S 1.2$ Normed Fields: Proposition $1.14$
Category:Characterisation of Non-Archimedean Division Ring Norms
Characterisation of Non-Archimedean Division Ring Norms/Corollary 4

== Theorem ==
Let $\struct {R, \norm {\,\cdot\,} }$ be a division ring with unity $1_R$.
Let $R$ have characteristic $p > 0$.
Then $\norm {\,\cdot\,}$ is a non_Archimedean norm on $R$.
== Proof ==
Because $R$ has characteristic $p > 0$, the set:
:$\set {n \cdot 1_k: n \in \Z}$
has cardinality $p - 1$.
Therefore:
:$\sup \set {\norm {n \cdot 1_R}: n \in \Z} = \max \set {\norm {1 \cdot 1_R}, \norm {2 \cdot 1_R}, \cdots, \norm {\paren {p - 1} \cdot 1_R} } < +\infty$
By Corollary 2:
:$\norm{\,\cdot\,}$ is non-Archimedean and $C = 1$.
Category:Characterisation of Non-Archimedean Division Ring Norms
Characterisation of Non-Archimedean Division Ring Norms/Corollary 5

== Theorem ==
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with unity $1_R$.
If $\norm {\, \cdot \,}$ is non-Archimedean then:
:$\sup \set {\norm {n \cdot 1_R}: n \in \Z} = 1$
where $n \cdot 1_R = 
\begin{cases}
\underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} } & : n > 0 \\
0 & : n = 0 \\
\\
-\underbrace {\paren {1_R + 1_R + \dots + 1_R} }_{\text {$-n$ times} } & : n < 0 \\
\end{cases}$
== Proof ==
By Corollary 1 of Characterisation of Non-Archimedean Division Ring Norms then:
:$\sup \set {\norm{n \cdot 1_R}: n \in \N_{> 0}} = 1$
By Norm Axiom $(\text N 1)$ (Positive Definiteness) then:
:$\norm {0 \cdot 1_R} = 0 \le 1$
Let $n < 0$ then:
The result follows.
== Sources ==
* : $\S 2.2$ Basic Properties, Theorem $2.2.2$
* : $\S 1.2$ Normed Fields, Proposition $1.14$
Category:Characterisation of Non-Archimedean Division Ring Norms
Characterisation of Non-Archimedean Division Ring Norms/Necessary Condition

== Theorem ==
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.
Then:
:$\norm {\,\cdot\,}$ is non-Archimedean $\implies \forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$.
where:
$n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} }$
== Proof ==
Let $\norm {\,\cdot\,}$ be non-Archimedean.
Then by the definition of a non-Archimedean norm, for $n \in \N$:
== Sources ==
* : $\S 1.2$ Normed Fields, Proposition $1.14$
* : $\S 2.2$ Basic Properties, Theorem $2.2.2$
Category:Characterisation of Non-Archimedean Division Ring Norms
Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition

== Theorem ==
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.
Then:
:$\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1 \implies \norm {\,\cdot\,}$ is non-archimedean
where:
:$n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{n \text { times} }$
== Proof ==
Let:
:$\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$
Let $x, y \in R$.
Let $y = 0_R$ where $0_R$ is the zero of $R$.
Then $\norm {x + y} = \norm x = \max \set {\norm x, 0} = \max \set {\norm x, \norm y}$
==== Lemma 1 ====
Hence to complete the proof it is sufficient to prove:
:$\forall x \in R: \norm {x + 1_R} \le \max \set {\norm x, 1}$
For $n \in \N$:
==== Lemma 2 ====
Hence
Taking $n$th roots yields:
:$\norm {x + 1_R} \le \paren {n + 1}^{1/n} \max \set {\norm x, 1}$
==== Lemma 3 ====
By the Multiple Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {n + 1}^{1/n} \max \set {\norm x, 1} = \max \set {\norm x, 1}$
By Inequality Rule for Real Sequences:
:$\norm {x + 1_R} \le \max \set {\norm x, 1}$
The result follows.
== Sources ==
* : $\S 1.2$ Normed Fields, Proposition $1.14$
* : $\S 2.2$ Basic Properties, Theorem $2.2.2$
Category:Characterisation of Non-Archimedean Division Ring Norms
Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 1

== Theorem ==
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.
Let $x, y \in R$.
Let $y \ne 0_R$ where $0_R$ is the zero of $R$.
Then:
:$\norm {x + y} \le \max \set {\norm x, \norm y} \iff \norm {x y^{-1} + 1_R} \le \max \set {\norm {x y^{-1} }, 1}$
== Proof ==
== Sources ==
* : $\S 1.2$ Normed Fields, Proposition $1.14$
* : $\S 2.2$ Basic Properties, Theorem $2.2.2$
Category:Characterisation of Non-Archimedean Division Ring Norms
Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 2

== Theorem ==
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.
Let $x \in R$.
Let $n \in \N$.
Then for all $i$, $0 \le i \le n$:
:$\norm x^i \le \max \set {\norm x^n , 1}$
== Proof ==
If $\norm x > 1$ then for all $i$, $0 \le i \le n$:
:$\norm x^i \le \norm x^n \le \max \set {\norm x^n, 1}$
If $\norm x \le 1$ then for all $i$, $0 \le i \le n$:
:$\norm x^i \le 1 \le \max \set {\norm x^n, 1}$
In either case for all $i$, $0 \le i \le n$:
:$\norm x^i \le \max \set {\norm x^n , 1}$
== Sources ==
* : $\S 1.2$ Normed Fields, Proposition $1.14$
* : $\S 2.2$ Basic Properties, Theorem $2.2.2$
Category:Characterisation of Non-Archimedean Division Ring Norms
Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 3

== Theorem ==
Let $\sequence {x_n}$ be the real sequence defined as $x_n = \paren {n + 1}^{1/n}$, using exponentiation.
Then $\sequence {x_n}$ converges with a limit of $1$.
== Proof ==
We have the definition of the power to a real number:
:$\paren {n + 1}^{1/n} = \map \exp {\dfrac 1 n \map \ln {n + 1} }$
For $n \ge 1$ then $n + 1 \le 2 n$.
Hence:
By Powers Drown Logarithms:
:$\ds \lim_{n \mathop \to \infty} \frac 1 n \ln n = 0$
By Sequence of Reciprocals is Null Sequence:
:$\ds \lim_{n \mathop \to \infty} \frac 1 n = 0$
By Combined Sum Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {\frac {\ln 2} n + \frac 1 n \ln n} = \ln 2 \cdot 0 + 0 = 0$
By the Squeeze Theorem for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {n + 1}^{1/n} = 0$
Hence:
:$\ds \lim_{n \mathop \to \infty} \paren {n + 1}^{1/n} = \exp 0 = 1$
and the result follows.
Category:Characterisation of Non-Archimedean Division Ring Norms
Characterisation of Real Symmetric Positive Definite Matrix

== Theorem ==
Let $A$ be an $n \times n$ symmetric matrix over $\mathbb R$.
Then $A$ is positive definite :
:there exists a nonsingular matrix $C$ such that $A = C^\intercal C$.
== Proof ==
=== Necessary Condition ===
=== Sufficient Condition ===
Category:Characterisation of Real Symmetric Positive Definite Matrix
Category:Symmetric Matrices
Category:Positive Definite Matrices
Characterisation of Real Symmetric Positive Definite Matrix/Necessary Condition

== Theorem ==
Let $A$ be an $n \times n$ positive definite symmetric matrix over $\RR$.
Then:
:there exists a nonsingular matrix $C$ such that $A = C^\intercal C$.
== Proof ==
Let $A$ be positive definite. 
From Real Symmetric Matrix is Orthogonally Diagonalizable:
:there exists an orthogonal matrix $P$ and diagonal matrix $D$ such that $A = P^\intercal D P$.
Further:
:the diagonal entries of $D$ are the eigenvalues of $A$. 
From Real Symmetric Positive Definite Matrix has Positive Eigenvalues:
:the diagonal entries of $D$ are positive. 
We can therefore construct a real diagonal matrix $S$ by: 
:$\paren S_{i j} = \begin{cases} \sqrt {\paren D_{i i} } & i = j \\ 0 & i \ne j \end{cases}$
From Product of Diagonal Matrices is Diagonal, we have: 
:$\paren {S^2}_{i j} = \begin{cases} \paren D_{i i} & i = j \\ 0 & i \ne j \end{cases}$
so:
:$S^2 = D$
We also have: 
We therefore have: 
So from Matrix is Nonsingular iff Determinant has Multiplicative Inverse:
:$P^\intercal S P$ is nonsingular.
Let $C = P^\intercal S P$. 
Then:
As $C$ is nonsingular, the proof is complete.
Category:Characterisation of Real Symmetric Positive Definite Matrix
Characterisation of Real Symmetric Positive Definite Matrix/Sufficient Condition

== Theorem ==
Let $A$ be an $n \times n$ symmetric matrix over $\mathbb R$ such that: 
:there exists a nonsingular matrix $C$ such that $A = C^\intercal C$.
Then $A$ is positive definite.
== Proof ==
Let $A$ be a symmetric matrix such that: 
:there exists an nonsingular matrix $C$ such that $A = C^\intercal C$.
Let $\mathbf v$ be a non-zero vector. 
Then:
So $A$ is positive definite. 
Category:Characterisation of Real Symmetric Positive Definite Matrix
Characterisation of Sine and Cosine

== Theorem ==
Let $\map s x: \R \to \R$, $\map c x: \R \to \R$ be differentiable real functions that satisfy:
:$(1): \quad \map {s'} x = \map c x$
:$(2): \quad \map {c'} x = -\map s x$
:$(3): \quad \map s 0 = 0$
:$(4): \quad \map c 0 = 1$
:$(5): \quad \forall x: \map {s^2} x + \map {c^2} x = 1$
Then, for every $x \in \R$:
:$\map s x = \map \sin x$
:$\map c x = \map \cos x$
== Proof ==
Define:
:$\map h x = \paren {\map c x - \map \cos x}^2 + \paren {\map s x - \map \sin x}^2$
Then:
By taking $\map {h'} x$:
By Zero Derivative implies Constant Function, $\map h x$ is a constant function:
:$\map h x = k$
Also:
Since $\map h x$ is constant:
:$\map h x = 0$
Therefore:
:$\paren {\map c x - \map \cos x}^2 + \paren {\map s x - \map \sin x}^2 = 0$
for every $x \in \R$.
But since Square of Real Number is Non-Negative and Square of Non-Zero Real Number is Strictly Positive:
:$\map c x - \map \cos x = 0$
:$\map s x - \map \sin x = 0$
The result follows.
Category:Sine Function
Category:Cosine Function
Characterisation of Spanning Set through Free Module Indexed by Set

== Theorem ==
Let $M$ be a unitary $R$-module.
Let $S = \family {m_i}_{i \mathop \in I}$ be a family of elements of $M$.
Let $\Psi: R^{\paren I} \to M$ be the morphism given by Universal Property of Free Module Indexed by Set.
Then $S$ is a spanning set of $M$ $\Psi$ is surjective.
== Proof ==
For $\family {r_i}_{i \mathop \in I} \in R^{\paren I}$ we have:
:$\map \Psi {\family {r_i}_{i \mathop \in I} } = \ds \sum_{i \mathop \in I} m_i r_i$
Thus $\Psi$ is surjective every element of $M$ is a linear combination of $S$.
Category:Module Theory
Characterisation of Terminal P-adic Expansion

== Theorem ==
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $x \in \Q_p$.
Then:
:the $p$-adic expansion of $x$ terminates 
 
:$\exists a \in \N : \exists k \in \Z : x = \dfrac a {p^k}$
== Proof ==
=== Necessary Condition ===
=== Sufficient Condition ===
== Sources ==
* : $\S 1.6$ The $p$-adic expansion of rational numbers: Exercise $34$
Category:P-adic Number Theory
Category:Characterisation of Terminal P-adic Expansion
Characterisation of Terminal P-adic Expansion/Necessary Condition

== Theorem ==
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $x \in \Q_p$.
Let the $p$-adic expansion of $x$ terminate.
Then:
:$\exists a \in \N : \exists k \in \Z : x = \dfrac a {p^k}$
== Proof ==
Let the $p$-adic expansion of $x$ be:
:$x = \ds \sum_{n \mathop = m}^\infty d_n p^n$ 
where:
:$m \in \Z_{\le 0}$
:$\forall n \in \Z_{\ge m}: d_n$ is a $p$-adic digit
:$m < 0 \implies d_m \ne 0$
By the definition of terminates:
:$\exists n_0 \in \N : n_0 \ge m : \forall n \ge n_0 : d_n = 0$
We have:
Let:
:$k = -m$
Let:
:$a = \ds \sum_{n \mathop = 0}^{n_0 - m} d_{n + m} p^n $
Then:
:$x = \dfrac a {p^k}$
The result follows.
Category:Characterisation of Terminal P-adic Expansion
Characterisation of Terminal P-adic Expansion/Sufficient Condition

== Theorem ==
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $a \in \N$.
Let $k \in \Z$.
Let $x = \dfrac a {p^k}$.
Then:
:the $p$-adic expansion of $x$ terminates
== Proof ==
From Basis Representation Theorem, $a$ can be expressed uniquely in the form:
:$\ds a = \sum_{j \mathop = 0}^n d_j p^j$
where:
:$n$ is such that $p^n \le a < p^{n + 1}$
:all the $d_j$ are such that $0 \le d_j < p$.
We have:
Let:
:$m = \begin{cases}
-k & : -k \le 0\\
0 & : 0 < -k 
\end{cases}$
For each $i : m \le i \le n-k$, let:
:$e_i = \begin{cases}
d_{i + k} & : -k \le i \le n-k\\
0 & : m \le i < -k
\end{cases}$
For each $i > n-k$, let:
:$e_i = 0$
Then:
:$x = \ds \sum_{i \mathop = m}^\infty e_i p^i$
where:
:$\forall i \ge m: 0 \le e_i < p$
:$\forall i > n-k: e_i = 0$
Hence $\ds \sum_{i \mathop = m}^\infty e_i p^i$ is a terminal $p$-adic expansion by definition.
From P-adic Expansion Representative of P-adic Number is Unique, the $p$-adic expansion of $x$ is:
:$x = \ds \sum_{i \mathop = m}^\infty e_i p^i$
Category:Characterisation of Terminal P-adic Expansion
Characteristic Function Determined by 0-Fiber

== Theorem ==
Let $A \subseteq S$.
Let $f: S \to \set {0, 1}$ be a mapping.
Denote by $\chi_A$ the characteristic function on $A$.
Then the following are equivalent:
:$(1): \quad f = \chi_A$
:$(2): \quad \forall s \in S: \map f s = 0 \iff s \notin A$
Using the notion of a fiber, $(2)$ may also be expressed as:
:$(2'):\quad \map {f^{-1} } 0 = S \setminus A$
== Proof ==
=== $(1)$ implies $(2)$ ===
Follows directly from the definition of characteristic function.
=== $(2)$ implies $(1)$ ===
Let $s \in S$.
Suppose that $s \notin A$.
Then by assumption, $\map f s = 0$.
Also, by definition of characteristic function, $\map {\chi_A} s = 0$.
Next, suppose that $s \in A$.
Then $\map f s \ne 0$ by assumption.
As $\map f s \in \set {0, 1}$, it follows that $\map f s = 1$.
Again, by definition of characteristic function, also have $\map {\chi_A} s = 1$.
Hence, for all $s \in S$, have $\map f s = \map {\chi_A} s$.
By Equality of Mappings, it follows that $f = \chi_A$.
Category:Characteristic Functions