nyuuzyou/wikis huggingface dataset:
Real Sequence/Examples/(-1)^(n+1) over n

== Examples of Real Sequence ==
The first few terms of the real sequence:
:$S = \sequence {\dfrac {\paren {-1}^{n + 1} } n}$
are:
:$1, -\dfrac 1 2, \dfrac 1 3, -\dfrac 1 4, \dotsc$
== Sources ==
* : Chapter $1$: Sequences: $1$. Infinite Sequences
Category:Examples of Real Sequences
Real Sequence/Examples/(-1)^n

== Examples of Real Sequence ==
The first few terms of the real sequence:
:$S = \sequence {\paren {-1}^n}_{n \mathop \ge 1}$
are:
:$-1, +1, -1, +1, \dotsc$
This is an example of the real sequence:
:$S = \sequence {x^n}$
where $x = -1$.
$S$ is not monotone, either increasing or decreasing.
== Sources ==
* : Chapter $1$: Sequences: $1$. Infinite Sequences: Example $4$
* : $\S 4$: Convergent Sequences: $\S 4.3$: Examples: $\text{(i)}$
* : $\S 4$: Convergent Sequences: $\S 4.16$: Example $\text{(iv)}$
* 
* 
Category:Examples of Real Sequences
Real Sequence/Examples/(-1)^n 2^(-n)

== Examples of Real Sequence ==
The real sequence whose first few terms are:
:$1, -\dfrac 1 2, \dfrac 1 4, -\dfrac 1 8, \dotsc$
can be defined by the formula:
:$S = \sequence {\paren {-1}^n \dfrac 1 {2^n} }_{n \mathop \ge 0}$
== Sources ==
* : $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Example $1.2.1 \, \text {(b)}$
Category:Examples of Real Sequences
Real Sequence/Examples/(1 + (-1)^n) / 2

== Examples of Real Sequence ==
The real sequence whose first few terms are:
:$1, 0, 1, 0, \dotsc$
can be defined by the formula:
:$S = \sequence {\dfrac {1 + \paren {-1}^n} 2}_{n \mathop \ge 0}$
== Sources ==
* : $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Example $1.2.1 \, \text {(e)}$
Category:Examples of Real Sequences
Real Sequence/Examples/1

== Examples of Real Sequence ==
The first few terms of the real sequence:
:$S = \sequence 1_{n \mathop \ge 1}$
are:
:$1, 1, 1, 1, \dotsc$
$S$ is both increasing and decreasing.
== Sources ==
* : Chapter $1$: Sequences: $1$. Infinite Sequences: Example $3$
* : $\S 4$: Convergent Sequences: $\S 4.3$: Examples: $\text{(iii)}$
* : $\S 4$: Convergent Sequences: $\S 4.16$: Example $\text{(iii)}$
Category:Examples of Real Sequences
Real Sequence/Examples/2^n

== Examples of Real Sequence ==
The first few terms of the real sequence:
:$S = \sequence {2^n}_{n \mathop \ge 1}$
are:
:$2, 4, 8, 16, \dotsc$
$S$ is strictly increasing.
== Sources ==
* : $\S 4$: Convergent Sequences: $\S 4.3$: Examples: $\text{(iv)}$
* : $\S 4$: Convergent Sequences: $\S 4.16$: Example $\text{(i)}$
Category:Examples of Real Sequences
Real Sequence/Examples/Arbitrary Sequence 1

== Examples of Real Sequence ==
The real sequence whose first few terms are:
:$\dfrac 1 2, 1, -\dfrac 1 2, -1, \dfrac 1 4, \dfrac 1 2, \dotsc$
has no obvious formula to define it.
== Discussion ==
The source work from which this example comes says:
:In these examples, there is a simple formula for $s_n$ in terms of $n$, which the reader will spot.
Neither the author of this page nor his tutors for this part of his degree course were able to identify what that "simple" rule is.
Note that it is straightforward to design a rule which, while not being simple as such, is sufficient to do the job, for example:
:$S = \sequence {\paren {-1} ^ {\ceiling {\frac n 2} - 1} \times 2 ^ {\ceiling {\frac n 4} - \paren {n - 1} \bmod 2} }$
or:
:$s_n = \begin {cases} \dfrac 1 2 & : n = 1 \\ \dfrac {s_{n - 1} } {1 - n} & : \text {$n$ odd} \\ 2 s_{n - 1} & : \text {$n$ even}, n > 0 \end {cases}$
== Sources ==
* : $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Example $1.2.1 \, \text {(c)}$
* 
Category:Examples of Real Sequences
Real Sequence/Examples/Arbitrary Sequence 2

== Examples of Real Sequence ==
The recurrence relation:
:$s_n = \begin {cases} 1 & : n = 1 \\ 0 & : n = 2 \\ \dfrac {s_{n - 2} + s_{n - 1} } 2 & : n > 2 \end {cases}$
defines a real sequence whose first few terms are:
:$1, 0, \dfrac 1 2, \dfrac 1 4, \dfrac 3 8, \dfrac 5 {16}, \dotsc$
== Sources ==
* : $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Example $1.2.1 \, \text {(f)}$
Category:Examples of Real Sequences
Real Sequence/Examples/Half of (x (n-1) + 2 over x (n-1))

== Examples of Real Sequence ==
The first few terms of the real sequence:
:$S = \sequence {a_n}_{n \mathop \ge 1}$
defined as:
:$a_n = \begin {cases} 2 & : n = 1 \\ \dfrac 1 2 \paren {x_{n - 1} + \dfrac 2 {x_{n - 1} } } & : n > 1 \end {cases}$
are:
:$2, \dfrac 3 2, \dfrac {17} {12}, \dfrac {577} {408}, \dotsc$
== Sources ==
* : $\S 4$: Convergent Sequences: $\S 4.3$: Examples: $\text{(v)}$
Category:Examples of Real Sequences
Real Sequence/Examples/Root (2 + Root x(n))

== Example of Real Sequence ==
Let $\sequence {x_n}$ denote the real sequence defined as:
:$x_n = \begin {cases} \sqrt 2 : n = 1 \\ \sqrt {2 + \sqrt {x_{n - 1} } } & : n > 1 \end {cases}$
Then $\sequence {x_n}$ converges to a root of $x^4 - 4 x^2 - x + 4 = 0$ between $\sqrt 3$ and $2$.
== Proof ==
:
Because $x_1 = \sqrt 2$, we have that:
:$x_2 > \sqrt 2 = x_1$
Suppose that:
:$x_n > x_n - 1$
for some $n \ge 2$.
Then:
Hence $\sequence {x_n}$ is seen to be strictly increasing.
Next we note that if $x_n > 1$:
As $x_n$ is strictly increasing, it follows that $x_n > \sqrt 3$ for all $n \ge 1$.
Similarly, we note that if $x_n < 2$:
So $\sequence {x_n} \to k$ where $\sqrt 3 < k \le 2$.
By definition of $x_n$:
By investigating the shape of the graph, it is seen that this is the only root of $x^4 - 4 x^2 - x + 4 = 0$ strictly greater than $1$.
The other root, as is seen by inspection, is in fact $1$.
== Sources ==
* : $1$: Review of some real analysis: Exercise $1.5: 14$
Category:Examples of Real Sequences
Real Sequence/Examples/n

== Examples of Real Sequence ==
The real sequence whose first few terms are:
:$1, 2, 3, \dotsc$
can be defined by the formula:
:$S = \sequence n_{n \mathop \ge 1}$
== Sources ==
* : Chapter $1$: Sequences: $1$. Infinite Sequences
* : $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Example $1.2.1 \, \text {(d)}$
Category:Examples of Real Sequences
Real Sequence/Examples/n^(-1)

== Examples of Real Sequence ==
The first few terms of the real sequence:
:$S = \sequence {n^{-1} }_{n \mathop \ge 1}$
are:
:$1, \dfrac 1 2, \dfrac 1 3, \dfrac 1 4, \dotsc$
$S$ is strictly decreasing.
== Sources ==
* : $\S 4$: Convergent Sequences: $\S 4.3$: Examples: $\text{(ii)}$
* : $\S 4$: Convergent Sequences: $\S 4.16$: Example $\text{(ii)}$
Category:Examples of Real Sequences
Real Sequence/Examples/n^s

== Examples of Real Sequence ==
Let $s$ be a constant.
The first few terms of the real sequence:
:$S = \sequence {n^s}$
are:
:$1^s, 2^s, 3^s, \ldots$
When $s = 1$, $S$ is the sequence of natural numbers.
== Sources ==
* : Chapter $1$: Sequences: $1$. Infinite Sequences: Example $2$
Category:Examples of Real Sequences
Real Sequence/Examples/n over (n+1)

== Examples of Real Sequence ==
The real sequence $S$ whose first few terms are:
:$\dfrac 1 2, \dfrac 2 3, \dfrac 3 4, \dotsc$
can be defined by the formula:
:$S = \sequence {\dfrac n {n + 1} }_{n \mathop \ge 1}$
$S$ is strictly increasing.
== Proof ==
Let $s_n$ denote the $n$th term of $S$.
We have:
Hence $S$ is increasing by definition.
== Sources ==
* : Chapter $\text I$. Continuity: $2$. Upper or Lower Limit of a Sequence
* : $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Example $1.2.1 \, \text {(a)}$
Category:Examples of Real Sequences
Real Sequence/Examples/x(n+1) = 2 over x(n)+1

== Example of Real Sequence ==
Let $\sequence {x_n}$ denote the real sequence defined as:
:$x_n = \begin {cases} a : 0 < a < 1 & : n = 1 \\ \dfrac 2 {x_n + 1} & : n > 1 \end {cases}$
Then the subsequences $\sequence {x_{2 n} }$ and $\sequence {x_{2 n + 1} }$ are both monotone:
:$\sequence {2 n}$ is strictly decreasing
:$\sequence {2 n + 1}$ is strictly increasing
Hence $\sequence {x_n} \to 1$ as $n \to \infty$.
== Proof ==
We are given that $0 < x_1 < 1$.
Then we have that:
:$\dfrac 2 {0 + 1} > x_2 > \dfrac 2 {1 + 1}$
That is:
:$1 < x_2 < 2$
Then we have that:
:$\dfrac 2 {1 + 1} > x_3 > \dfrac 2 {2 + 1}$
That is:
:$\dfrac 2 3 < x_3 < 1$
and so certainly:
:$0 < x_3 < 1$
Hence in general:
:$0 < x_n < 1 \implies 1 < x_{n + 1} < 2$
and:
:$1 < x_n < 2 \implies 0 < x_{n + 1} < 1$
So:
:for $n$ odd, $0 < x_n < 1$
:for $n$ even, $1 < x_n < 2$
Now we show:
We calculate the difference between $x_n$ and $x_{n + 2}$:
and we see that:
:if $0 < x_n < 1$ then $x_n - x_{n + 2} < 0$
:if $1 < x_n < 2$ then $x_n - x_{n + 2} > 0$
Hence it follows that:
:for $n$ odd, $\sequence {2 n + 1}$ is strictly increasing
:for $n$ even, $\sequence {2 n}$ is strictly decreasing
and the result follows from Convergence of Odd and Even Subsequences to Same Limit.
== Sources ==
* : $1$: Review of some real analysis: Exercise $1.5: 13$
Category:Examples of Real Sequences
Real Sequence/Examples/x^n over n Factorial

== Examples of Real Sequence ==
The real sequence whose first few terms are:
:$\dfrac x {1!}, \dfrac {x^2} {2!}, \dfrac {x^3} {3!}, \dfrac {x^4} {4!}, \dotsc$
can be defined by the formula:
:$S = \sequence {\dfrac {x^n} {n!} }_{n \mathop \ge 1}$
== Sources ==
* 
* 
Category:Examples of Real Sequences
Real Sequence (1 + x over n)^n is Convergent

== Theorem ==
The sequence $\sequence {s_n}$ defined as:
:$s_n = \paren {1 + \dfrac x n}^n$
is convergent.
== Proof ==
From Cauchy's Mean Theorem:
:$(1): \quad \ds \paren {\prod_{k \mathop = 1}^n a_k}^{1/n} \le \frac 1 n \paren {\sum_{k \mathop = 1}^n a_k}$
for $r_1, r_2, \ldots, r_n$.
Setting:
:$a_1 = a_2 = \ldots = a_{n - 1} := 1 + \dfrac x {n - 1}$
and:
:$a_n = 1$
Substituting for $a_1, a_2, \ldots, a_n$ into $(1)$ gives:
The above is valid only if $a_1, a_2, \ldots, a_n$ are positive.
Hence we have shown that $\sequence {s_n}$ is increasing when:
:$1 + \dfrac x {n - 1} \ge 0$
that is, when:
:$n \ge 1 - x$
From Equivalence of Definitions of Real Exponential Function: Limit of Sequence implies Power Series Expansion, we have:
:$(2): \quad \paren {1 + \dfrac x n}^n \le 1 + \size x + \dfrac {\size x^2} {2!} + \dotsb + \dfrac {\size x^n} {n!}$
Since there exists $N$ such that:
:$\forall n > N: \dfrac {\size x^n} {n!} \le \paren {\dfrac 1 2}^n$
it follows from $(2)$ that:
Hence we have that $\sequence {s_n}$ is strictly increasing and bounded above.
So by the Monotone Convergence Theorem (Real Analysis), $\sequence {s_n}$ is convergent.
As $1 + \dfrac x n$ is positive when $n$ is large enough, it follows that the limit of $\sequence {s_n}$ is positive.
== Sources ==
* : $\S 5$: Subsequences: Exercise $\S 5.7 \ (5)$
Category:Exponential Function
Real Sequence with Nonzero Limit is Eventually Nonzero

== Theorem ==
Let $\sequence {x_n}$ be a real sequence.
Let $\sequence {x_n}$ converge to $a \ne 0$.
Then:
:$\exists N \in \N: \forall n \ge N: x_n \ne 0$
That is, eventually every term of $\sequence {x_n}$ becomes non-zero.
== Proof ==
Suppose $a > 0$. 
By Sequence Converges to Within Half Limit:
:$\exists N \in \N: \forall n > N: x_n > \dfrac a 2 > 0$
Now suppose $a < 0$. 
By Sequence Converges to Within Half Limit:
:$\exists N \in \N: \forall n > N: x_n < \dfrac a 2 < 0$
This shows that if $a \ne 0$:
:$\exists N \in \N: \forall n > N: x_n \ne 0$
== Also see ==
* Limit of Positive Real Sequence is Positive
Category:Convergence
Real Sequence with all Subsequences having Convergent Subsequence to Limit Converges to Same Limit

== Theorem ==
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence. 
Let $x \in \R$. 
Suppose that:
:every subsequence $\sequence {x_{n_j} }_{j \in \N}$ of $\sequence {x_n}_{n \in \mathop \N}$ has a subsequence $\sequence {x_{n_{j_k} } }_{k \in \N}$ such that: 
::$x_{n_{j_k} } \to x$ 
Then: 
:$x_n \to x$
== Proof ==
, suppose that: 
:$x_n$ does not converge to $x$.
Then, there exists some $\epsilon > 0$ such that for every $k \in \N$ there exists $n_k \ge k$ such that: 
:$\size {x_{n_k} - x} \ge \epsilon$
Let $\sequence {x_{n_{j_k} } }_{k \in \N}$ be a subsequence of $\sequence {x_{n_j} }_{j \in \N}$.
Then: 
:$\size {x_{n_{j_k} } - x} \ge \epsilon$ 
for each $k \in \N$.
In particular, there exists no $N \in \N$ such that: 
:$\size {x_{n_{j_k} } - x} < \epsilon$
for each $k \ge N$, so: 
:$\sequence {x_{n_{j_k} } }_{k \mathop \in \N}$ does not converge to $x$.
So:
:$\sequence {x_{n_j} }_{j \mathop \in \N}$ is a subsequence of $\sequence {x_n}_{n \mathop \in \N}$, but has a subsequence that does not converge to $x$.
This contradicts our hypothesis on the subsequences of $\sequence {x_n}_{n \mathop \in \N}$. 
So $x_n$ converges to $x$.
Category:Limits of Sequences
Real Sine Function is Bounded

== Theorem ==
Let $x \in \R$.
Then:
:$\size {\sin x} \le 1$
== Proof ==
From the algebraic definition of the real sine function:
:$\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$
it follows that $\sin x$ is a real function.
Thus $\sin^2 x \ge 0$.
From Sum of Squares of Sine and Cosine‎, we have that $\cos^2 x + \sin^2 x = 1$.
Thus it follows that:
:$\sin^2 x = 1 - \cos^2 x \le 1$
From Ordering of Squares in Reals and the definition of absolute value, we have that:
:$x^2 \le 1 \iff \size x \le 1$
The result follows.
== Also see ==
* Complex Sine Function is Unbounded
== Sources ==
* : $\S 16.3 \ (3) \ \text{(ii)}$
Category:Sine Function
Real Sine Function is Continuous

== Theorem ==
The real sine function $\sin: \R \to \R$ is continuous on $\R$.
== Proof ==
Recall the definition of the sine function:
:$\ds \forall x \in \R: \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$
Thus $\sin x$ is expressible in the form of a power series.
From Sine Function is Absolutely Convergent, we have that the interval of convergence of $\sin x$ is the whole of $\R$. 
From Power Series is Differentiable on Interval of Convergence, it follows that $\sin x$ is continuous on the whole of $\R$.
Category:Sine Function
Real Sine Function is neither Injective nor Surjective

== Theorem ==
The real sine function is neither an injection nor a surjection.
== Proof ==
This is immediately apparent from the graph of the sine function:
For example:
:$\map \sin 0 = \map \sin \pi = 0$
and so the real sine function is not an injection.
Then, for example:
:$\nexists x \in \R: \map \sin x = 2$
and so the real sine function is not a surjection.
== Sources ==
* : Chapter $2$: Maps and relations on sets: Example $2.5$
Category:Sine Function
Category:Examples of Injections
Category:Examples of Surjections
Real Square Function is neither Injective nor Surjective

== Example of Mapping which is Not an Injection nor a Surjection ==
Let $f: \R \to \R$ be the real square function:
:$\forall x \in \R: \map f x = x^2$
Then $f$ is neither an injection nor a surjection.
== Proof ==
Consider the real square function:
:
From Real Square Function is not Injective:
:$f$ is not an injection.
From Real Square Function is not Surjective:
:$f$ is not a surjection.
== Sources ==
* : $\S 5$. Induced mappings; composition; injections; surjections; bijections: Exercise $6$
Category:Examples of Injections
Category:Examples of Surjections
Category:Square Function
Real Square Function is not Bijective

== Theorem ==
Let $f: \R \to \R$ be the real square function:
:$\forall x \in \R: \map f x = x^2$
Then $f$ is not a bijection.
== Proof ==
From Real Square Function is not Injective, $f$ is not an injection.
From Real Square Function is not Surjective, $f$ is not a surjection.
The result follows by definition of bijection.
== Sources ==
* : Chapter $3$. Mappings: Exercise $3$
Category:Square Function
Real Square Function is not Injective

== Example of Mapping which is Not an Injection ==
Let $f: \R \to \R$ be the real square function:
:$\forall x \in \R: \map f x = x^2$
Then $f$ is not an injection.
== Proof ==
For $f$ to be an injection, it would be necessary that:
:$\forall x_1, x_2 \in \R: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$
By definition of the squaring operation, we have:
:$\map f x = \map f {-x}$
But unless $x = 0$ it is not the case that $x = -x$.
Hence $f$ is not an injection.
== Sources ==
* : $\S 22$: Injections; bijections; inverse of a bijection
Category:Examples of Injections
Category:Square Function
Real Square Function is not Surjective

== Example of Mapping which is Not a Surjection ==
Let $f: \R \to \R$ be the real square function:
:$\forall x \in \R: \map f x = x^2$
Then $f$ is not a surjection.
== Proof ==
For $f$ to be a surjection, it would be necessary that:
:$\forall y \in \R: \exists x \in \R: \map f x = y$
However from Square of Real Number is Non-Negative:
:$\forall y \in \R_{< 0}: \nexists x \in \R: \map f x = y$
Hence $f$ is not a surjection.
Category:Examples of Surjections
Category:Square Function
Real Star-Algebra is Commutative

== Theorem ==
Let $A = \struct {A_F, \oplus}$ be a real $*$-algebra whose conjugation is denoted as $*$.
Then:
:$\forall a, b \in A, a \oplus b = b \oplus a$
That is, real $*$-algebra is commutative.
== Proof ==
Category:Real Star-Algebras
Category:Commutative Algebras
Real Subtraction is Closed

== Theorem ==
The set of real numbers is closed under subtraction:
:$\forall a, b \in \R: a - b \in \R$
== Proof ==
From the definition of real subtraction:
:$a - b := a + \paren {-b}$
where $-b$ is the inverse for real number addition.
From Real Numbers under Addition form Group:
:$\forall a, b \in \R: a + \paren {-b} \in \R$
Therefore real number subtraction is closed.
== Sources ==
* : Chapter $1$: The Real and Complex Number Systems: $\text{1-2}$ Arithmetical properties of real numbers: Axiom $4$
* : Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$
* : Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Example $1$
Category:Real Subtraction
Category:Algebraic Closure
Real Symmetric Matrix is Hermitian

== Theorem ==
Every real symmetric matrix is Hermitian.
== Proof ==
Let $\mathbf A$ be a real symmetric matrix.
Then we have:
So:
:$\mathbf A = \mathbf A^\dagger$
Thus, by definition, $\mathbf A$ is Hermitian.
== Sources ==
* 
Category:Symmetric Matrices
Category:Hermitian Matrices
Real Symmetric Positive Definite Matrix has Positive Eigenvalues

== Theorem ==
Let $A$ be a symmetric positive definite matrix over $\mathbb R$.
Let $\lambda$ be an eigenvalue of $A$. 
Then $\lambda$ is real with $\lambda > 0$.
 
== Proof ==
Let $\lambda$ be an eigenvalue of $A$ and let $\mathbf v$ be a corresponding eigenvector.
From Real Symmetric Matrix has Real Eigenvalues, $\lambda$ is real.
From the definition of a positive definite matrix, we have: 
:$\mathbf v^\intercal A \mathbf v > 0$
That is: 
From Euclidean Space is Normed Vector Space, we have: 
:$\norm {\mathbf v}^2 > 0$
so:
:$\lambda > 0$
Category:Symmetric Matrices
Category:Positive Definite Matrices
Real Trigonometric Functions are Continuous

== Theorem ==
The trigonometric functions on the real numbers $\R$:
:sine
:cosine
:tangent
:cotangent
:secant
:cosecant
are all continuous at every point of their domain.
== Proof ==
In turn:
:Real Sine Function is Continuous
:Real Cosine Function is Continuous
:Real Tangent Function is Continuous
:Real Cotangent Function is Continuous
:Real Secant Function is Continuous
:Real Cosecant Function is Continuous
== Sources ==
* 
* 
Category:Continuous Real Functions
Category:Trigonometric Functions
Real Vector Space under Chebyshev Distance is Homeomorphic to that under Euclidean Metric

== Theorem ==
Let $\R^n$ be an $n$-dimensional real vector space.
Let $d_\infty: \R^n \times \R^n \to \R$ be the Chebyshev distance on $\R^n$.
Let $d_2: \R^n \times \R^n \to \R$ be the Euclidean metric on $\R^n$.
Let $M_1 = \struct {\R^n, d_\infty}$ and $M_2 = \struct {\R^n, d_2}$ be the corresponding metric spaces.
Then $M_1$ and $M_2$ are homeomorphic.
== Proof ==
From Relation between $p$-Product Metric and Chebyshev Distance on Real Vector Space:
:$\forall x, y \in \R^n: \map {d_\infty} {x, y} \le \map {d_p} {x, y} \le n^{1 / p} \map {d_\infty} {x, y}$
The Euclidean metric $d_2$ is a special case of the $p$-product metric $d_p$ for $p = 2$.
It follows by definition that $d_\infty$ and $d_2$ are Lipschitz equivalent.
It follows from Lipschitz Equivalent Metrics are Topologically Equivalent that $d_\infty$ and $d_2$ are topologically equivalent.
The result follows from Metric Spaces on Topologically Equivalent Metrics on same Underlying Set are Homeomorphic.
== Sources ==
* : Chapter $2$: Metric Spaces: $\S 7$: Subspaces and Equivalence of Metric Spaces
Category:Homeomorphisms (Metric Spaces)
Category:Chebyshev Distance
Category:Euclidean Metric
Real Vector Space with Dot Product is Hilbert Space

== Theorem ==
Let $\R^d$ be a real vector space with $d$ dimensions.
Let $\innerprod \cdot \cdot$ be the dot product on $\R^d$.
Then $\R^d$ endowed with $\innerprod \cdot \cdot$ is a Hilbert space.
== Proof ==
Let us explore the inner product norm of $\innerprod \cdot \cdot$:
:$\ds \norm x = \sqrt{ \sum_{i \mathop = 1}^d x_i^2 }$
and subsequently the metric induced by $\norm \cdot$:
:$\ds \map d {x, y} = \norm {x - y} = \sqrt{ \sum_{i \mathop = 1}^d \paren{ x_i - y_i }^2 }$
which is seen to be the definition of the Euclidean metric.
By Euclidean Space is Complete Metric Space, it follows that $\R^d$ is complete with respect to $d$.
That is, $\R^d$ is a Hilbert space.
== Sources ==
* : $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples
Category:Examples of Hilbert Spaces
Real Zero is Zero Element

== Theorem ==
:$\forall x \in \R: 0 \times x = 0$
== Proof ==
== Sources ==
* : Chapter $1$: Real Numbers and Functions of a Real Variable: $1.2$ Operations with Real Numbers: $(7)$
* : $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(b)}$
Category:Real Numbers
Category:Zero
Real and Imaginary Part Projections are Continuous

== Theorem ==
Define the real-valued functions $x, y: \C \to \R$ by:
:$\forall z \in \C: \map x z = \map \Re z$
:$\forall z \in \C: \map y z = \map \Im z$
Equip $\R$ with the usual Euclidean metric.
Equip $\C$ with the usual Euclidean metric.
Then both $x$ and $y$ are continuous functions.
== Proof ==
Let $z \in \C$, and let $\epsilon \in \R_{>0}$.
Put $\delta = \epsilon$.
For all $w \in \C$ with $\cmod {w - z} < \delta$:
and
It follows by definition that $x$ and $y$ are both continuous.
== Notes ==
This theorem can be considered a special case of Continuous Mapping to Product Space.
Suppose we let $z = \map \Re z + i \map \Im z$ be a complex number.
We can now identify the complex number $z$ with the ordered pair $\tuple {\map \Re z, \map \Im z} \in \R^2$, where $R^2$ is the Cartesian product $\R \times \R$.
The functions $x$ and $y$ can now be considered as projections on the co-ordinates.
Category:Complex Numbers
Category:Continuous Functions
Real and Imaginary Parts of Complex-Valued Function Vanishing at Infinity Vanish at Infinity

== Theorem ==
Let $X$ be a locally compact Hausdorff space.
Let $f : X \to \C$ be a complex-valued function vanishing at infinity.
Then $\map \Re f$ and $\map \Im f$ vanish at infinity.
== Proof ==
Let $\epsilon > 0$. 
Since $f$ vanishes at infinity, there exists a compact $K \subseteq X$ such that:
:$\cmod {\map f x}^2 < \epsilon^2$ for each $x \in X \setminus K$. 
Then we have:
:$\paren {\map \Re {\map f x} }^2 + \paren {\map \Im {\map f x} }^2 < \epsilon^2$ 
So that:
:$\paren {\map \Re {\map f x} }^2 < \epsilon^2$
and:
:$\paren {\map \Im {\map f x} }^2 < \epsilon^2$
for each $x \in X \setminus K$. 
Then:
:$\cmod {\map \Re {\map f x} } < \epsilon$
and:
:$\cmod {\map \Im {\map f x} } < \epsilon$
for each $x \in X \setminus K$. 
Hence both $\map \Re f$ and $\map \Im f$ vanish at infinity.
Category:Complex-Valued Functions Vanishing at Infinity