hcju/proofwikips huggingface dataset:
Half-Range Fourier Sine Series/Sine of Non-Integer Multiple of x over 0 to Pi
Let $\lambda \in \R \setminus \Z$ be a real number which is not an integer. Let $\map f x$ be the real function defined on $\openint 0 \pi$ as: :$\map f x = \sin \lambda x$ Then its half-range Fourier sine series can be expressed as: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \frac {2 \sin \lambda \pi} \pi \paren {\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {n \sin n x} {\lambda^2 - n^2} }       | c =  }} {{eqn | r = \frac {2 \sin \lambda \pi} \pi \paren {-\frac {\sin x} {\lambda^2 - 1} + \frac {2 \sin 2 x} {\lambda^2 - 4} - \frac {3 \sin 3 x} {\lambda^2 - 9}  + \frac {4 \sin 4 x} {\lambda^2 - 16} - \dotsb}       | c =  }} {{end-eqn}}
Minimum Rule for Continuous Functions
Let $\struct {S, \tau}$ be a topological space. Let $f, g: S \to \R$ be continuous real-valued functions. Let $\min \set {f, g}: S \to \R$ denote the pointwise minimum of $f$ and $g$. Then: :$\min \set {f, g}$ is continuous.
Min is Half of Sum Less Absolute Difference
For all numbers $a, b$ where $a, b$ in $\N, \Z, \Q$ or $\R$: :$\min \set {a, b} = \dfrac 1 2 \paren {a + b - \size {a - b} }$
Continuity Test for Real-Valued Functions
Let $\struct{S, \tau}$ be a topological space. Let $f: S \to \R$ be a real-valued function. Let $x \in S$. Then $f$ is continuous at $x$ : :$\forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$
Continuity Test for Real-Valued Functions/Everywhere Continuous
Let $\struct{S, \tau}$ be a topological space. Let $f: S \to \R$ be a real-valued function. Then $f$ is everywhere continuous : :$\forall x \in S : \forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$
Oesterlé-Masser Conjecture
Let $\epsilon \in \R$ be a strictly positive real number.
Oesterlé-Masser Conjecture/Formulation 1
There exists only a finite number of triples of (strictly) positive integers $\tuple {a, b, c}$ with the conditions: :$a + b = c$ :$a$, $b$ and $c$ are pairwise coprime such that: :$c > \map {\operatorname {rad} } {a b c}^{1 + \epsilon}$ where $\operatorname {rad}$ denotes the radical of an integer.
Definite Integral of Function satisfying Dirichlet Conditions is Continuous
Let $f: \R \to \R$ be a real function defined in the open interval $\openint {-\pi} \pi$. Let $f$ fulfil the Dirichlet conditions in $\openint {-\pi} \pi$. Let $a_0, a_1, \dotsc; b_1, \dotsc$ be the Fourier coefficients of $f$ in $\openint {-\pi} \pi$. Then the real function: :$\map F x = \displaystyle \int_{-\pi}^x \map f t \rd t - \dfrac {a_0} 2 x$ is continuous on $\openint {-\pi} \pi$.
Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 1
Let $G := \set {x \in \R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$. Let $\circ: G \times G \to G$ be the binary operation defined as: :$\forall x, y \in G: x \circ y = \dfrac {x + y} {1 + x y}$ {{:Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers}}
Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 2
Let $G := \set {x \in \R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$. Let $\circ: G \times G \to G$ be the binary operation defined as: :$\forall x, y \in G: x \circ y = \dfrac {x + y} {1 + x y}$ {{:Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers}}
Triangle Inequality/Complex Numbers/Corollary 2
Let $z_1, z_2 \in \C$ be complex numbers. Let $\cmod z$ be the modulus of $z$. Then: :$\cmod {z_1 + z_2} \ge \cmod {\cmod {z_1} - \cmod {z_2} }$
Continuity Test for Real-Valued Functions/Necessary Condition
Let $\struct {S, \tau}$ be a topological space. Let $f: S \to \R$ be a real-valued function. Let $x \in S$. Let $f$ be continuous at $x$ Then: :$\forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$
Continuity Test for Real-Valued Functions/Sufficient Condition
Let $\struct {S, \tau}$ be a topological space. Let $f: S \to \R$ be a real-valued function. Let $x \in S$. Let $f$ satisfy: :$\forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$ Then $f$ is continuous at $x$
Equivalence of Definitions of Completely Hausdorff Space
{{TFAE|def = Completely Hausdorff Space|view = a completely Hausdorff space}} Let $T = \struct {S, \tau}$ be a topological space.
Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff/Necessary Condition
Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct {S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Let $T$ be a completely Hausdorff space. Then for each $\alpha \in I$, $\struct {S_\alpha, \tau_\alpha}$ is a completely Hausdorff space.
Metric Space is T4
Let $M = \struct {A, d}$ be a metric space. Then $M$ is $T_4$.
Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff/Sufficient Condition
Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$. Let each of $\struct {S_\alpha, \tau_\alpha}$ for $\alpha \in I$ be completely Hausdorff spaces. Then $T$ is a completely Hausdorff spaces.
P-adic Expansion Less Intial Zero Terms Represents Same P-adic Number
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\mathbf a$ be an equivalence class in $\Q_p$. Let $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$ be a $p$-adic expansion that represents $\mathbf a$. Let $l$ be the first index $i \ge m$ such that $d_i \ne 0$ Then the series: :$\displaystyle \sum_{i \mathop = l}^\infty d_i p^i$ also represents $\mathbf a$.
Subsequence is Equivalent to Cauchy Sequence
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ be a Cauchy sequence in $R$. Let $\sequence {x_{m_n} }$ be a subsequence of $\sequence {x_n}$. Then: :$\displaystyle \lim_{n \mathop \to \infty} {x_n - x_{m_n} } = 0$
Numbers with Square-Free Binomial Coefficients/Lemma
Let $n$ be a (strictly) positive integer.  Let $p$ be a prime number. By Basis Representation Theorem, there is a unique sequence $\sequence {a_j}_{0 \mathop \le j \mathop \le r}$ such that: :$(1): \quad \displaystyle n = \sum_{k \mathop = 0}^r a_k p^k$ :$(2): \quad \displaystyle \forall k \in \closedint 0 r: a_k \in \N_b$ :$(3): \quad r_t \ne 0$ Suppose $r \ge 2$ and $p^2 \nmid \dbinom n m$ for all $0 \le m \le n$.  Then: :$p^{r - 1} \divides \paren {n + 1}$ that is: :$p^{r - 1}$ divides $\paren {n + 1}$.
Derivative of Generating Function/General Result/Corollary
Let the coefficient of $z^n$ extracted from $\map G z$ be denoted: :$\sqbrk {z^n} \map G z := a_n$ Then: :$\sqbrk {z^m} \map G z = \dfrac 1 {m!} \map {G^{\paren m} } 0$ where $G^{\paren m}$ denotes the $m$th derivative of $G$.
Derivative of Generating Function/General Result
Let $m$ be a positive integer. Then: :$\dfrac {\d^m} {\d z^m} \map G z = \displaystyle \sum_{k \mathop \ge 0} \dfrac {\paren {k + m}!} {k!} a_{k + m} z^k$ === Corollary === {{:Derivative of Generating Function/General Result/Corollary}}
Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm form Banach Space
Let $I = \closedint a b$ be a closed real interval. Let $\map \CC I$ be the space of real-valued functions, continuous on $I$. Let $\norm {\,\cdot\,}_\infty$ be the supremum norm on real-valued functions, continuous on $I$. Then $\struct {\map \CC I, \norm {\,\cdot\,}_\infty}$ is a Banach space.
Inductive Set under Progressing Mapping has Minimally Inductive Subset
Let $A$ be an inductive class under a mapping $g$. Let $A$ be a set. Then there exists some subset $S$ of $A$ such that $S$ is minimally inductive under $g$.
Recurrence Relation for Bell Numbers
Let $B_n$ be the Bell number for $n \in \Z_{\ge 0}$. Then: :$B_{n + 1} = \displaystyle \sum_{k \mathop = 0}^n \dbinom n k B_k$ where $\dbinom n k$ are binomial coefficients.
Natural Number m is Less than n implies n is not Greater than Successor of n
Let $\N$ be the natural numbers. Let $m, n \in \N$. Then: :$m < n \implies m + 1 \le n$
Natural Number Ordering is Preserved by Successor Mapping
Let $\N$ be the natural numbers. Let $m, n \in \N$. Then: :$n \le m \implies n^+ \le m^+$
Non-Empty Bounded Subset of Natural Numbers has Greatest Element
Let $\omega$ be the set of natural numbers defined as the von Neumann construction. Then every non-empty bounded subset of $\omega$ has a greatest element.
Natural Number Less than or Equal to Successor of Another
Let $\N$ be the natural numbers. Let $m, n \in \N$ such that $m \le n^+$. Then either: :$(1): \quad m \le n$ or: :$(2): \quad m = n^+$
Mapping whose Image of Natural Number n is Subset of Image of Successor
Let $f: \N \to A$ be a mapping from the set of natural numbers $\N$ to a class $A$. Let $f$ have the property that: :$\forall n \in \N: \map f n \subseteq \map f {n^+}$ where $n^+$ is the successor of $n$. Then: :$\forall n, m \in N: n \le m \implies \map f n \subseteq \map f m$
Mapping whose Image of Natural Number n is Subset of Image of Successor/Corollary
Let $f$ have the property that: :$\forall n \in \N: \map f n \subsetneq \map f {n^+}$ where $n^+$ is the successor of $n$. Then: :$\forall n, m \in N: n < m \implies \map f n \subsetneq \map f m$
Non-Empty Finite Set of Natural Numbers has Greatest Element
Let $A$ be a non-empty finite set of natural numbers. Then $A$ has a greatest element.
Set of Subsets of Element of Minimally Inductive Class is Finite
Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Let $x \in M$. Let $S$ be the set of all $y \in M$ such that $y \subseteq x$. Then $S$ is finite.
Minimally Inductive Class with Fixed Element is Finite
Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Let there exist an element $x \in M$ such that $x = \map g x$. Then $M$ is a finite class.
Set of Natural Numbers is either Finite or Denumerable
Let $S$ be a subset of the natural numbers $\N$. Then $S$ is either finite or denumerable.
Unlike Electric Charges Attract
Let $a$ and $b$ be stationary particles, each carrying an electric charge of $q_a$ and $q_b$ respectively. Let $q_a$ and $q_b$ be of the opposite sign. That is, let $q_a$ and $q_b$ be unlike charges. Then the forces exerted by $a$ on $b$, and by $b$ on $a$, are such as to cause $a$ and $b$ to attract each other.
Value of Vacuum Permittivity
The value of the '''vacuum permittivity''' is calculated as: :$\varepsilon_0 = 8 \cdotp 85418 \, 78128 (13) \times 10^{-12} \, \mathrm F \, \mathrm m^{-1}$ (farads per metre) with a relative uncertainty of $1 \cdotp 5 \times 10^{-10}$.
Successor Mapping on Natural Numbers has no Fixed Element
Let $\N$ denote the set of natural numbers. Then: :$\forall n \in \N: n + 1 \ne n$
Residue of Fibonacci Number Modulo Fibonacci Number/Lemma
:$F_{m n + 1} \equiv \paren {\begin{cases} F_1 & : m \bmod 4 = 0 \\ F_{n - 1} & : m \bmod 4 = 1 \\ \paren {-1}^n F_1 & : m \bmod 4 = 2 \\ \paren {-1}^n F_{n - 1} & : m \bmod 4 = 3 \end{cases} } \pmod {F_n}$
Complement of Interval Defined by Absolute Value
Let $\xi, \delta \in \R$ be real numbers. Let $\delta > 0$. Then:
Complement of Open Interval Defined by Absolute Value
:$\set {x \in \R: \size {\xi - x} \ge \delta} = \R \setminus \openint {\xi - \delta} {\xi + \delta}$ where: :$\openint {\xi - \delta} {\xi + \delta}$ is the open real interval between $\xi - \delta$ and $\xi + \delta$ :$\setminus$ denotes the set difference operator.
Complement of Closed Interval Defined by Absolute Value
:$\set {x \in \R: \size {\xi - x} > \delta} = \R \setminus \closedint {\xi - \delta} {\xi + \delta}$ where: :$\closedint {\xi - \delta} {\xi + \delta}$ is the closed real interval between $\xi - \delta$ and $\xi + \delta$ :$\setminus$ denotes the set difference operator.
Size of y-1 lt n and Size of y+1 gt 1 over n
Let $T_n \subseteq \R$ be the subset of the set of real numbers $\R$ defined as: :$T_n = \set {y: \size {y - 1} < n \land \size {y + 1} > \dfrac 1 n}$ Then: :$T_n = \openint {1 - n} {-1 - \dfrac 1 n} \cup \openint {-1 + \dfrac 1 n} {1 + n}$
Real Part as Mapping is Surjection
Let $f: \C \to \R$ be the projection from the complex numbers to the real numbers defined as: :$\forall z \in \C: \map f z = \map \Re z$ where $\map \Re z$ denotes the real part of $z$. Then $f$ is a surjection.
Imaginary Part as Mapping is Surjection
Let $f: \C \to \R$ be the projection from the complex numbers to the real numbers defined as: :$\forall z \in \C: \map f z = \map \Im z$ where $\map \Im z$ denotes the imaginary part of $z$. Then $f$ is a surjection.
Condition for Mapping from Quotient Set to be Well-Defined
Let $S$ and $T$ be sets. Let $\RR$ be an equivalence relation on $S$. Let $f: S \to T$ be a mapping from $S$ to $T$. Let $S / \RR$ be the quotient set of $S$ induced by $\RR$. Let $q_\RR: S \to S / \RR$ be the quotient mapping induced by $\RR$. Then: :there exists a mapping $\phi: S / \RR \to T$ such that $\phi \circ q_\RR = f$ : :$\forall x, y \in S: \tuple {x, y} \in \RR \implies \map f x = \map f y$ ::$\begin {xy} \xymatrix@L + 2mu@ + 1em {  S \ar[r]^*{f}    \ar[d]_*{q_\RR} & T \\ S / \RR \ar@{-->}[ur]_*{\phi} } \end {xy}$
Condition for Mapping from Quotient Set to be Surjection
Let the mapping $\phi: S / \RR \to T$ defined as: :$\phi \circ q_\RR = f$ be well-defined. Then: :$\phi$ is a surjection : :$f$ is a surjection.
Condition for Mapping from Quotient Set to be Injection
Let the mapping $\phi: S / \RR \to T$ defined as: :$\phi \circ q_\RR = f$ be well-defined. Then: :$\phi$ is an injection : :$\forall x, y \in S: \tuple {x, y} \in \RR \iff \map f x = \map f y$
Mapping from Quotient Set when Defined is Unique
Let the mapping $\phi: S / \RR \to T$ defined as: :$\phi \circ q_\RR = f$ be well-defined. Then $\phi$ is unique.
Conditions for Commutative Diagram on Quotient Mappings between Mappings
Let $A$ and $B$ be sets. Let $\RR_S$ and $\RR_T$ be equivalence relations on $S$ and $T$ respectively. Let $f: S \to T$ be a mapping from $S$ to $T$. Let $S / \RR_S$ and $T / \RR_T$ be the quotient sets of $S$ and $T$ induced by $\RR_S$ and $\RR_T$ respectively. Let $q_S: S \to S / \RR_S$ and $q_T: T \to T / \RR_T$ be the quotient mappings induced by $\RR_S$ and $\RR_T$ respectively. Then a mapping $g: S / \RR_S \to T / \RR_T$ exists such that: :$q_T \circ f = g \circ q_S$ : :$\forall x, y \in S: x \mathrel {\RR_S} y \implies \map f x \mathrel {\RR_T} \map f y$ ::$\begin {xy} \xymatrix@L + 2mu@ + 1em {  S \ar[r]^*{f}    \ar[d]_*{q_S} & T \ar[d]^*{q_T} \\ S / \RR_S \ar@{-->}[r]_*{g} & T / \RR_T } \end {xy}$
Subset of Finite Dimensional Normed Vector Space is Compact iff Closed and Bounded/Sufficient Condition
Let $\struct {X, \norm {\,\cdot\,}}$ be a finite-dimensional normed vector space. Let $K \subset X$ be a compact subset. Then $K$ is closed and bounded.
Factors of Group Direct Product are not Subgroups
Let $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ be groups. Let $\struct {G \times H, \circ}$ be the group direct product of $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$. Then neither $\struct {G, \circ_1}$ nor $\struct {H, \circ_2}$ is a subgroup of $\struct {G \times H, \circ}$.
Additive Group of Complex Numbers is Direct Product of Reals with Reals
Let $\struct {\C, +}$ be the additive group of complex numbers. Let $\struct {\R, +}$ be the additive group of real numbers. Then the direct product $\struct {\R, +} \times \struct {\R, +}$ is isomorphic with $\struct {\C, +}$.
Imaginary Numbers under Addition form Group
Let $\II$ denote the set of complex numbers of the form $0 + i y$ That is, let $\II$ be the set of all wholly imaginary numbers. Then the algebraic structure $\struct {\II, +}$ is a group.
Imaginary Numbers under Multiplication do not form Group
Let $\II$ denote the set of complex numbers of the form $0 + i y$ for $y \in \R_{\ne 0}$. That is, let $\II$ be the set of all wholly imaginary non-zero numbers. Then the algebraic structure $\struct {\II, \times}$ is not a group.
Set of Isometries in Complex Plane under Composition forms Group
Let $S$ be the set of all complex functions $f: \C \to \C$ which preserve distance when embedded in the complex plane. That is: :$\size {\map f a - \map f b} = \size {a - b}$ Let $\struct {S, \circ}$ be the algebraic structure formed from $S$ and the composition operation $\circ$. Then $\struct {S, \circ}$ is a group.
Set of Affine Mappings on Real Line under Composition forms Group
Let $S$ be the set of all real functions $f: \R \to \R$ of the form: :$\forall x \in \R: \map f x = r x + s$ where $r \in \R_{\ne 0}$ and $s \in \R$ Let $\struct {S, \circ}$ be the algebraic structure formed from $S$ and the composition operation $\circ$. Then $\struct {S, \circ}$ is a group.
Arbitrary Cyclic Group of Order 4
Let $S = \set {1, 2, 3, 4}$. Consider the algebraic structure $\struct {S, \circ}$ given by the Cayley table: :$\begin{array}{r|rrrr} \circ & 2 & 3 & 4 & 1 \\ \hline 2 & 2 & 3 & 4 & 1  \\ 3 & 3 & 4 & 1 & 2 \\ 4 & 4 & 1 & 2 & 3 \\ 1 & 1 & 2 & 3 & 4 \\ \end{array}$ Then $\struct {S, \circ}$ is a group. Specifically, $\struct {S, \circ}$ is the cyclic group of order $4$.
Factors of Sums of Powers of 100,000/General Result
All integers $n$ of the form: :$n = \displaystyle \sum_{k \mathop = 0}^m 10^{r k}$ for $m \in \Z_{> 0}$ are composite for $r \ge 2$. The only exceptions are $r = 2^k, m = 1$ for some $k \in \N$, and $r = m + 1 =$ some odd prime, where $n$ could be prime.
Odd Integers under Addition do not form Subgroup of Integers
Let $S$ denote the set of odd integers. Then $\struct {S, +}$ is not a subgroup of the additive group of integers $\struct {\Z, +}$.
Equivalence of Definitions of Closed Set in Normed Vector Space
{{TFAE|def = Closed Set in Normed Vector Space|view = Closed Set|context = Normed Vector Space|contextview = Normed Vector Spaces}} Let $V = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $F \subseteq X$.
Cosets of Positive Reals in Multiplicative Group of Complex Numbers
Let $S$ be the positive real axis of the complex plane: :$S = \set {z \in \C: z = x + 0 i, x \in \R_{>0} }$ Consider the algebraic structure $\struct {S, \times}$ as a subgroup of the multiplicative group of complex numbers $\struct {\C_{\ne 0}, \times}$. The cosets of $\struct {S, \times}$ are the sets of the form: :$\set {z \in \C: \exists r \in \R_{>0}: z = r e^{i \theta}}$ for some $\theta \in \hointr 0 {2 \pi}$ That is, the sets of all complex numbers with a constant argument.
Morphism from Multiplicative Group of Complex Numbers to Unit Circle
Let $\struct {\C_{\ne 0}, \times}$ denote the multiplicative group of complex numbers. Let $f: \C_{\ne 0} \to \C_{\ne 0}$ be the mapping defined as: :$\forall z \in \C_{\ne 0}: \map f z = \dfrac z {\cmod z}$ where $\cmod z$ denotes the modulus of $z$. Then $f$ is an endomorphism on $\struct {\C_{\ne 0}, \times}$ whose kernel is the positive real axis: :$\set {z \in \C: z = x + 0 i, x \in \R_{>0} }$ and whose image is the unit circle: :$\set {z \in \C: \cmod z = 1}$
Leigh.Samphier/Sandbox/P-adic Expansion Representative of P-adic Number is Unique
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where: :$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$ :$\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$. Let $\mathbf a$ be an equivalence class in $\Q_p$. Let $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$ and $\displaystyle \sum_{i \mathop = k}^\infty e_i p^i$ be $p$-adic expansions that represent $\mathbf a$. Then: :$(1) \quad m = k$ :$(2) \quad \forall i \ge m : d_i = e_i$ That is, the $p$-adic expansions $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$ and $\displaystyle \sum_{i \mathop = k}^\infty e_i p^i$ are identical.
Rational Numbers are Dense Subfield of P-adic Numbers
Let $p$ be a prime number. Let $\norm {\,\cdot\,}_p$ be the p-adic norm on the rationals $\Q$. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where: :$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$ :$\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$. Let $\phi: \Q \to \Q_p$ be the mapping defined by: :$\map \phi r = \sequence {r, r, r, \dotsc} + \NN$ where $\sequence {r, r, r, \dotsc} + \NN$ is the left coset in $\CC \, \big / \NN$ that contains the constant sequence $\sequence {r, r, r, \dotsc}$. Then: :$\Q$ is isometrically isomorphic to $\map \phi \Q$ which is a dense subfield of $\Q_p$.
Graph is 0-Regular iff Edgeless
Let $G$ be a graph. Then $G$ is $0$-regular graph  $G$ is edgeless.
Edgeless Graph of Order n has n Components
Let $N_n$ denote the edgeless graph with $n$ vertices. Then $N_n$ has $n$ components.
Complete Graph of Order 1 is Edgeless
The complete graph $K_1$ of order $1$ is the edgeless graph $N_1$.
Edgeless Graph of Order 1 is Tree
Let $N_1$ denote the edgeless graph with $1$ vertex. Then $N_1$ is a tree.
Edgeless Graph of Order Greater than 1 is Forest
Let $N_n$ denote the edgeless graph with $n$ vertices such that $n > 1$. Then $N_n$ is a forest.
Size of Complete Graph
Let $K_n$ denote the complete graph of order $n$ where $n \ge 0$. The size of $K_n$ is given by: :$\size {K_n} = \dfrac {n \paren {n - 1} } 2$
Field Operations of P-adic Numbers as Quotient of Cauchy Sequences
Let $p$ be a prime number. Let $\norm {\,\cdot\,}_p$ be the p-adic norm on the rationals $\Q$. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where: :$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$ :$\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$. Then the field operations on $\Q_p$ are defined by: :$+ : \quad \forall \sequence {x_n} + \NN, \sequence {y_n} + \NN \in \CC \, \big / \NN$: :::$\quad \paren {\sequence {x_n} + \NN} + \paren {\sequence {y_n} + \NN} = \sequence {x_n + y_n} + \NN$ :$\circ : \quad \forall \sequence {x_n} + \NN, \sequence {y_n} + \NN \in \CC \, \big / \NN$:  :::$\quad \paren {\sequence {x_n} + \NN} \paren {\sequence {y_n} + \NN} = \sequence {x_n y_n} + \NN$
Size of Graph/Examples/Order 3 Graphs
Let $G$ be a simple graph of order $3$. Then it is possible for $G$ to have a size of $0$, $1$, $2$ or $3$. Examples of each are presented below: :600px
Simple Graph where All Vertices and All Edges are Adjacent
Let $G$ be a simple graph in which: :every vertex is adjacent to every other vertex and: :every edge is adjacent to every other edge. Then $G$ is of order no greater than $3$.
Simple Graph whose Vertices all Incident but Edges not Adjacent
Let $G = \struct {V, E}$ be a simple graph such that: :every vertex is incident with at least one edge :no two edges are adjacent to each other. Then $G$ has an even number of vertices.
Size of Star Graph
Let $S_n$ denote the star graph of order $n$ where $n > 0$. The size of $S_n$ is given by: :$\size {S_n} = n - 1$
Smallest Simple Graph with One Vertex Adjacent to All Others
Let $G = \struct {V, E}$ be a simple graph of order $n$. Let $G$ be the simple graph with the smallest size such that one vertex is adjacent to all other vertices of $G$. Then $G$ is the star graph of order $n$ and is of size $n - 1$.
Heine-Borel Theorem/Normed Vector Space
Let $\struct {X, \norm {\,\cdot\,}}$ be a finite-dimensional normed vector space. A subset $K \subset X$ is compact  $K$ is closed and bounded.