hcju/proofwikips huggingface dataset:
Definition:Ring of Square Matrices
Let $R$ be a ring. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\map {\mathcal M_R} n$ denote the $n \times n$ matrix space over $R$. Let $+$ denote the operation of matrix entrywise addition. Let $\times$ be (temporarily) used to denote the operation of conventional matrix multiplication.  The algebraic structure: :$\struct {\map {\mathcal M_R} n, +, \times}$ is known as the '''ring of square matrices of order $n$ over $R$'''
Definition:Additive Group
=== Additive Group of Integers === {{:Definition:Additive Group of Integers}} === Additive Group of Integer Multiples === {{:Definition:Additive Group of Integer Multiples}} === Additive Group of Integers Modulo $m$ === {{:Definition:Additive Group of Integers Modulo m}} === Additive Group of Rational Numbers === {{:Definition:Additive Group of Rational Numbers}} === Additive Group of Real Numbers === {{:Definition:Additive Group of Real Numbers}} === Additive Group of Complex Numbers === {{:Definition:Additive Group of Complex Numbers}}
Definition:Excess Kurtosis
'''Excess kurtosis''' is defined as the difference between the kurtosis of a particular probability distribution and that of the Gaussian distribution. Let $X$ be a random variable with kurtosis $\alpha_4$.  By Kurtosis of Gaussian Distribution, the kurtosis of a Gaussian distribution is equal to $3$.  So the '''excess kurtosis''' of $X$, usually denoted $\gamma_2$, is given by:  :$\gamma_2 = \alpha_4 - 3$
Definition:Die/Historical Note
Dice have been around for thousands of years. An early reference to the design of a die can be found in {{AuthorRef|W.R. Paton}}'s $1918$ translation of {{BookLink|The Greek Anthology Book XIV|W.R. Paton}}: :''The numbers on a die run so: six one, five two, three four.''
Definition:Ordered Integral Domain/Definition 1
An '''ordered integral domain''' is an integral domain $\struct {D, +, \times}$ which has a strict positivity property $P$: {{:Definition:Strict Positivity Property}}
Definition:Ordered Integral Domain/Definition 2
An '''ordered integral domain''' is an ordered ring $\struct {D, +, \times, \le}$ which is also an integral domain. That is, it is an integral domain with an ordering $\le$ compatible with the ring structure of $\struct {D, +, \times}$: {{begin-axiom}} {{axiom | n = \text {OID} 1         | lc= $\le$ is compatible with ring addition:         | q = \forall a, b, c \in D         | ml= a \le b         | mo= \implies         | mr= \paren {a + c} \le \paren {b + c} }} {{axiom | n = \text {OID} 2         | lc= Strict positivity is closed under ring product:         | q = \forall a, b \in D         | ml= 0_D \le a, 0_D \le b         | mo= \implies         | mr= 0_D \le a \times b }} {{end-axiom}}
Definition:Positivity Property
{{begin-axiom}} {{axiom | n = \text P 1         | m = P + P \subseteq P }} {{axiom | n = \text P 2         | m = P \cap \paren {-P} = \set {0_R} }} {{axiom | n = \text P 3         | m = P \circ P \subseteq P }} {{end-axiom}}
Definition:Total Ordering Induced by Strict Positivity Property
Let $\struct {D, +, \times, \le}$ be an ordered integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $P: D \to \set {\mathrm T, \mathrm F}$ denote the strict positivity property: {{:Definition:Strict Positivity Property}} Then the total ordering $\le$ compatible with the ring structure of $D$ is called the '''(total) ordering induced by (the strict positivity property) $P$'''.
Definition:Dottie Number
The '''Dottie number''' $\d$ is defined as the unique fixed point of the cosine function: :$\cos \rd = \d$
Definition:Strict Ordering on Integers
=== Definition 1 === {{:Definition:Strict Ordering on Integers/Definition 1}} === Definition 2 === {{:Definition:Strict Ordering on Integers/Definition 2}}
Definition:Well-Ordered Integral Domain/Definition 1
$\struct {D, +, \times \le}$ is a '''well-ordered integral domain'''  the ordering $\le$ is a well-ordering on the set $P$ of (strictly) positive elements of $D$.
Definition:Well-Ordered Integral Domain/Definition 2
$\struct {D, +, \times \le}$ is a '''well-ordered integral domain'''  every subset $S$ of the set $P$ of (strictly) positive elements of $D$ has a minimal element: :$\forall S \subseteq D_{\ge 0_D}: \exists x \in S: \forall a \in S: x \le a$ where $D_{\ge 0_D}$ denotes all the elements $d \in D$ such that $\map P d$.
Definition:P-adic Number/P-adic Norm Completion of Rational Numbers
The '''p-adic numbers''', denoted $\struct {\Q_p, \norm {\,\cdot\,}_p}$, is the unique (up to isometric isomorphism) non-Archimedean valued field that completes $\struct {\Q, \norm {\,\cdot\,}_p}$. Each element of $\Q_p$ is called a '''$p$-adic number'''.
Definition:P-adic Number/Quotient of Cauchy Sequences in P-adic Norm
Let $\CC$ be the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$. Let $\NN$ be the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$. Let $\Q_p$ denote the quotient ring $\CC \, \big / \NN$. Let $\norm {\, \cdot \,}_p:\Q_p \to \R_{\ge 0}$ be the norm on the quotient ring $\Q_p$ defined by: :$\displaystyle \forall \sequence {x_n} + \NN: \norm {\sequence {x_n} + \NN }_p = \lim_{n \mathop \to \infty} \norm{x_n}_p$ By Corollary to Quotient Ring of Cauchy Sequences is Normed Division Ring, then $\struct {\Q_p, \norm {\, \cdot \,}_p}$ is a valued field. Each left coset $\sequence {x_n} + \NN \in \CC \, \big / \NN$ is called a '''$p$-adic number'''. The '''p-adic numbers''' is the valued field $\struct {\Q_p, \norm {\,\cdot\,}_p}$.
Definition:Multiple/Integral Domain
Let $\struct {D, +, \circ}$ be an integral domain. Let $x, y \in D$. Let $x$ be a divisor of $y$. Then $y$ is a '''multiple''' of $x$.
Definition:Multiple/Integer
Let $\Z$ denote the set of integers. Let $x, y \in \Z$. Let $x$ be a divisor of $y$. Then $y$ is a '''multiple''' of $x$.
Definition:Unit of Ring/Definition 1
An element $x \in R$ is a '''unit of $\struct {R, +, \circ}$'''  $x$ is invertible under $\circ$. That is, a '''unit''' of $R$ is an element of $R$ which has an inverse. :$\exists y \in R: x \circ y = 1_R = y \circ x$
Definition:Unit of Ring/Definition 2
An element $x \in R$ is a '''unit of $\struct {R, +, \circ}$'''  $x$ is divisor of $1_R$.
Definition:Proper Divisor/Integer
Let $\struct {\Z, +, \times}$ be the ring of integers. Let $x, y \in \Z$. Then '''$x$ divides $y$''' is defined as: :$x \divides y \iff \exists t \in \Z: y = t \times x$ Then $x$ is a '''proper divisor''' of $y$ : :$(1): \quad x \divides y$ :$(2): \quad \size x \ne \size y$ :$(3): \quad x \ne \pm 1$ That is: :$(1): \quad x$ is a divisor of $y$ :$(2): \quad x$ and $y$ are not equal in absolute value :$(3): \quad x$ is not equal to either $1$ or $-1$.
Definition:Content of Polynomial/Integer
Let $f \in \Z \sqbrk X$ be a polynomial with integer coefficients. Then the '''content''' of $f$, denoted $\cont f$, is the greatest common divisor of the coefficients of $f$.
Definition:Content of Polynomial/Rational
Let $f \in \Q \sqbrk X$ be a polynomial with rational coefficients. The '''content''' of $f$ is defined as: :$\cont f := \dfrac {\cont {n f} } n$ where $n \in \N$ is such that $n f \in \Z \sqbrk X$.
Definition:Content of Polynomial/GCD Domain
Let $D$ be a GCD domain. Let $K$ be the field of quotients of $D$. Let $f \in K \sqbrk X$ be a polynomial. Let $a \in D$ be such that $a f \in D \sqbrk X$. Let $d$ be the greatest common divisor of the coefficients of $a f$. Then we define the '''content''' of $f$ to be: :$\cont f := \dfrac d a$
Definition:Z-Module Associated with Abelian Group/Definition 1
The '''$\Z$-module associated with $G$''' is the $\Z$-module $\struct {G, *, \circ}$ with ring action: :$\circ: \Z \times G \to G$: :$\tuple {n, x} \mapsto *^n x$ where $*^n x$ is the $n$th power of $x$.
Definition:Z-Module Associated with Abelian Group/Definition 2
The '''$\Z$-module associated with $G$''' is the $\Z$-module on $G$ with ring representation $\Z \to \map {\operatorname {End} } G$ equal to the initial homomorphism.
Definition:Linear Ring Action/Left
A '''(left) linear ring action''' of $R$ on $M$ is a mapping from the cartesian product $\circ : R \times M \to M$ such that: {{begin-axiom}} {{axiom | n = 1         | q = \forall \lambda \in R: \forall m, n \in M         | ml= \lambda \circ \paren {m + n}         | mo= =         | mr= \paren {\lambda \circ m} + \paren {\lambda \circ n} }} {{axiom | n = 2         | q = \forall \lambda, \mu \in R: \forall m \in M         | ml= \paren {\lambda + \mu} \circ m         | mo= =         | mr= \paren {\lambda \circ m} + \paren {\mu \circ m} }} {{axiom | n = 3         | q = \forall \lambda, \mu \in R: \forall m \in M         | ml= \paren {\lambda \mu} \circ m         | mo= =         | mr= \lambda \circ \paren {\mu \circ m} }} {{end-axiom}}
Definition:Linear Ring Action/Right
A '''right linear ring action''' of $R$ on $M$ is a mapping from the cartesian product $\circ : M \times R \to M$ such that: {{begin-axiom}} {{axiom | n = 1         | q = \forall \lambda \in R: \forall m, n \in M         | ml= \paren {m + n} \circ \lambda         | mo= =         | mr= \paren {m \circ \lambda} + \paren {n \circ \lambda} }} {{axiom | n = 2         | q = \forall \lambda, \mu \in R: \forall m \in M         | ml= m \circ \paren {\lambda + \mu}         | mo= =         | mr= \paren {m \circ \lambda} + \paren {m \circ \mu} }} {{axiom | n = 3         | q = \forall \lambda, \mu \in R: \forall m \in M         | ml= m \circ \paren {\lambda\mu}         | mo= =         | mr= \paren {m \circ \lambda} \circ \mu }} {{end-axiom}}
Definition:Closed Statement
Let $P$ be a statement. $P$ is a '''closed statement'''  $P$ contains only bound occurrences of any variables that may appear in it. That is, such that it contains no free occurrences of variables.
Definition:Open Statement
Let $P$ be a statement. $P$ is an '''open statement'''  $P$ contains at least one free occurrence of a variables that appears in it.
Definition:Generated Subspace
=== Definition 1 === {{:Definition:Generated Subspace/Definition 1}} === Definition 2 === {{:Definition:Generated Subspace/Definition 2}}
Definition:Generated Subspace/Definition 1
The '''subspace generated by $S$''' is the intersection of all subspaces of $\mathbf V$ containing $S$.
Definition:Generated Subspace/Definition 2
The '''subspace generated by $S$''' is the set of all linear combinations of elements of $S$.
Definition:Dimension of Vector Space/Definition 2
The '''dimension of $V$''' is the maximum cardinality of a linearly independent subset of $V$.
Definition:Dimension of Vector Space/Definition 1
The '''dimension of $V$''' is the number of vectors in a basis for $V$.
Definition:Sheldon Prime
A '''Sheldon prime''' is a prime number which satisfies both the product property and the mirror property.
Definition:Product Property of Primes
Let $p_n$ denote the $n$th prime number. $p_n$ satisfies the '''product property''' if the product of the digits of its base $10$ representation equals $n$.
Definition:Mirror Property of Primes
Let $p_n$ denote the $n$th prime number. Let $\map {\operatorname {rev} } {p_n}$ denote the reversal of $p_n$. $p_n$ satisfies the '''mirror property''' : :$(1): \quad \map {\operatorname {rev} } {p_n}$ is also a prime number :$(2): \quad$ the product of the digits of the base $10$ representation of $\map {\operatorname {rev} } {p_n}$ equals $\map {\operatorname {rev} } n$.
Definition:Smallest Field containing Subfield and Complex Number
Let $F$ be a field. Let $\theta \in \C$ be a complex number. Let $S$ be the intersection of all fields such that: :$S \subseteq F$ :$\theta \in F$ Then $S$ is denoted $\map F \theta$ and referred to as the '''smallest field containing $F$ and $\theta$'''. {{help|I encounter this definition in {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham}} without having much idea of the context. The element $\theta$ is specifically defined as being a complex number. At the moment I have not got a clue as to why the specific nature of $\theta$, but I will go along with it as defined in Clapham. If the context needs to be expanded, that can happen in due course.}} === General Definition === {{:Definition:Smallest Field containing Subfield and Complex Number/General Definition}}
Definition:Smallest Field containing Subfield and Complex Number/General Definition
Let $F$ be a field. Let $\theta_1, \theta_2, \ldots, \theta_n \in \C$ be complex numbers. Let $S$ be the intersection of all fields such that: :$S \subseteq F$ :$\theta_1, \theta_2, \ldots, \theta_n \in F$ Then $S$ is denoted $\map F {\theta_1, \theta_2, \ldots, \theta_n}$ and referred to as the '''smallest field containing $F$ and $\theta_1, \theta_2, \ldots, \theta_n$'''.
Definition:Algebraic Number over Field
Let $F$ be a field. Let $z$ be a complex number. $z$ is '''algebraic over $F$'''  $z$ is a root of a polynomial with coefficients in $F$.
Definition:Generated Field Extension/Definition 1
The '''field extension $F \sqbrk S$ generated by $S$''' is the smallest subfield extension of $E$ containing $S$, that is, the intersection of all subfields of $E$ containing $S$ and $F$. Thus $S$ is a '''generator''' of $F \sqbrk S$  $F \sqbrk S$ has no proper subfield extension containing $S$.
Definition:Simple Algebraic Field Extension
Let $F$ be a field. Let $z$ be a complex number which is algebraic over $F$. Then the smallest field $\map F z$ containing $F$ and $z$ is known as a '''simple algebraic (field) extension''' of $F$.
Definition:Algebraic Number over Field/Degree
Let $F$ be a field. Let $z \in \C$ be algebraic over $F$. The '''degree''' of $\alpha$ is the degree of the minimal polynomial $\map m x$ whose coefficients are all in $F$.
Definition:Algebraic Element of Field Extension/Definition 1
$\alpha$ is '''algebraic over $F$'''  it is a root of some nonzero polynomial over $F$: :$\exists f \in F \sqbrk X \setminus \set 0: \map f \alpha = 0$ where $F \sqbrk X$ denotes the ring of polynomial forms in $X$.
Definition:Algebraic Element of Field Extension/Definition 2
$\alpha$ is '''algebraic over $F$'''  the evaluation homomorphism $F \sqbrk X \to K$ at $\alpha$ is not injective.
Definition:Polynomial Evaluation Homomorphism/Single Indeterminate
Let $\struct {R \sqbrk X, \iota, X}$ be a polynomial ring in one variable over $R$. Let $s\in S$. A ring homomorphism $h : R \sqbrk X \to S$ is called an '''evaluation in $s$''' : :$\map h X = s$ :$h \circ \iota = \kappa$ where $\circ$ denotes composition of mappings.
Definition:Polynomial Evaluation Homomorphism/Multiple Indeterminates
Let $\family {s_i}_{i \mathop \in I}$ be an indexed family of elements of $S$. Let $R \sqbrk {\set {X_i: i \mathop \in I} }$ be a polynomial ring over $R$. A ring homomorphism $g: R \sqbrk {\set {X_i: i \mathop \in I} } \to S$ is called an '''evaluation at $\family {s_i}_{i \mathop \in I}$''' : :$\forall r \in R : \map g r = \map \kappa r$ :$\forall j \in J : \map g {X_j} = s_j$
Definition:Constructible Point in Plane
Let $\mathcal C$ be a Cartesian coordinate plane. Let $S$ be a set of points in $\mathcal C$. Let $P$ be a point in $\mathcal C$. Let there exist a compass and straightedge construction for $P$ from a line segment $AB$, where $A, B \in S$ Then $P$ is defined as '''constructible from $S$'''.
Definition:Coherent Sequence
Let $p$ be a prime number. Let $\sequence {\alpha_n}$ be an integer sequence such that: :$(1): \quad \forall n \in \N: 0 \le \alpha_n \le p^{n + 1} - 1$ :$(2): \quad \forall n \in \N: \alpha_{n + 1} \equiv \alpha_n \pmod {p^{n + 1}}$ The sequence $\sequence {\alpha_n}$ is said to be a '''coherent sequence'''. If it is necessary to emphasize the choice of prime $p$ then the sequence $\sequence {\alpha_n}$ is said to be a '''$p$-adically coherent sequence'''.
Definition:P-adic Norm/P-adic Numbers/P-adic Metric
The '''$p$-adic metric''' on $\Q_p$ is the metric induced by $\norm{\cdot}_p$: :$\forall x, y \in \Q_p: \map d {x, y} = \norm{x - y}_p$
Definition:P-adic Norm/P-adic Numbers
The norm $\norm {\,\cdot\,}_p$ on $\Q_p$ is called the '''$p$-adic norm''' on $\Q_p$.
Definition:Irreducible Element of Ring/Definition 1
$x$ is defined as '''irreducible'''  it has no non-trivial factorization in $D$. That is,  $x$ cannot be written as a product of two non-units.
Definition:Irreducible Element of Ring/Definition 2
$x$ is defined as '''irreducible'''  the only divisors of $x$ are its associates and the units of $D$. That is,  $x$ has no proper divisors.
Definition:Laplace Transform/Restriction to Reals
Although the definition of the '''Laplace transform''' has $s$ be a complex variable, sometimes the restriction of $\map {\laptrans f} s$ to wholly real $s$ is sufficient to solve a particular differential equation.  Therefore, elementary textbooks introducing the Laplace transform will often write something like the following: :''... where we assume at present that the parameter $s$ is real. Later it will be found useful to consider $s$ complex.'' ::::-- {{BookReference|Theory and Problems of Laplace Transforms|1965|Murray R. Spiegel}}: Chapter $1$: The Laplace Transform: Definition of the Laplace Transform :''A profound understanding of the workings of the Laplace transform requires considering it to be a so-called analytic function of a complex variable, but in most of this book we shall assume that the variable $s$ is real.'' ::::-- {{BookReference|Fourier Analysis and its Applications|2003|Anders Vretblad}}: $\S 3.1$
Definition:Laplace Transform/Graphical Interpretation
Define $\gamma$ as the integrand of $\displaystyle \int_0^{\to +\infty} e^{-s t} \map f t \rd t$ as a function of $\map f t$, $s$, and $t$: :$\map \gamma {\map f t, t; s} = \map f t \, e^{-s t}$ For any particular function $f$, holding $s$ fixed, the integrand of the Laplace Transform $\kappa$ can be interpreted as a contour. That is, for a given function $f$ and a particular complex number $s_0$ held constant: :$\map \kappa t: \R_{\ge 0} \to \C$ :$\map \kappa t = \map \gamma {\map f t, t; s_0}$ is a parameterization of a contour.
Definition:Laplace Transform/Discontinuity at Zero
Let $f: \R_{> 0} \to \mathbb F$ be a function of a real variable $t$, where $\mathbb F \in \set {\R, \C}$. Let $f$ be discontinuous or not defined at $t = 0$. Then the '''Laplace transform''' of $f$  is defined as: :$\displaystyle \laptrans {\map f t} = \map F s = \int_{0^+}^{\to +\infty} e^{-s t} \map f t \rd t = \lim_{\epsilon \mathop \to 0^+} \int_\epsilon^{\to +\infty} e^{-s t} \map f t \rd t$ whenever this improper integral converges. If this improper integral does not converge, then $\laptrans {\map f t}$ does not exist.
Definition:Cauchy Principal Value/Real Integral
Let $f: \R \to \R$ be a real function which is piecewise continuous everywhere. Then the '''Cauchy principal value of $\displaystyle \int f$''' is defined as: :$\PV_{-\infty}^{+\infty} \map f t \rd t := \lim_{R \mathop \to +\infty} \int_{-R}^R \map f t \rd t$ where $\displaystyle \int_{-R}^R \map f t \rd t$ is a Riemann integral.
Definition:Jump Discontinuity/Jump
Let $X$ be an open subset of $\R$. Let $f: X \to Y$ be a real function. Let $f$ be discontinuous at some point $c \in X$ such that $c$ is a '''jump discontinuity''' of $f$. The '''jump''' at $c$ is defined as: :$\displaystyle \lim_{x \mathop \to c^+} \map f x - \lim_{x \mathop \to c^-} \map f x$
Definition:Bessel Function/Order
The parameter $n$ is known as the '''order''' of the Bessel function.
Definition:Modified Bessel Function
The '''modified Bessel functions''' are solutions to Bessel's modified equation: :$x^2 \dfrac {\d^2 y} {\d x^2} + x \dfrac {\d y} {\d x} - \paren {x^2 + n^2} y = 0$ These solutions have two main classes: :the modified Bessel functions of the first kind $I_n$ and: :the modified Bessel functions of the second kind $K_n$.
Definition:Modified Bessel Function/Order
The parameter $n$ is known as the '''order''' of the modified Bessel function.
Definition:Modified Bessel Function/First Kind
A '''modified Bessel function of the first kind of order $n$''' is a modified Bessel function which is non-singular at the origin. It is usually denoted $\map {I_n} x$, where $x$ is the dependent variable of the instance of '''Bessel's modified equation''' to which $\map {I_n} x$ forms a solution.
Definition:Modified Bessel Function/Second Kind
A '''modified Bessel function of the second kind of order $n$''' is a modified Bessel function which is singular at the origin. It is usually denoted $\map {K_n} x$, where $x$ is the dependent variable of the instance of '''Bessel's modified equation''' to which $\map {K_n} x$ forms a solution.
Definition:Error Function
The '''error function''' is the following improper integral, considered as a real function $\erf : \R \to \R$: :$\map {\erf} x = \displaystyle \dfrac 2 {\sqrt \pi} \int_0^x \map \exp {-t^2} \rd t$ where $\exp$ is the real exponential function.
Definition:Complementary Error Function
The '''complementary error function''' is the real function $\erfc: \R \to \R$: {{begin-eqn}} {{eqn | l = \map {\erfc} x       | r = 1 - \map \erf x       | c = where $\erf$ denotes the Error Function }} {{eqn | r = 1 - \dfrac 2 {\sqrt \pi} \int_0^x \map \exp {-t^2} \rd t       | c = where $\exp$ denotes the Real Exponential Function }} {{eqn | r = \dfrac 2 {\sqrt \pi} \int_x^\infty \map \exp {-t^2} \rd t       | c =  }} {{end-eqn}}
Definition:Exponential Integral Function
The '''exponential integral function''' is the real function $\Ei: \R_{> 0} \to \R$ defined as: :$\map \Ei x = \displaystyle \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Definition:Null Function
Let $\mathcal N: \R \to \R$ be a real function such that: :$\forall x \in \R_{>0}: \displaystyle \int_0^x \map {\mathcal N} t \rd t = 0$ Then $\mathcal N$ is a '''null function'''.
Definition:Half Wave Rectified Sine Curve
The half wave rectified sine curve is the real function $f: \R \to \R$ defined as: :$\forall t \in \R: \map f t = \begin {cases} \sin t & : 2 n \pi \le t \le \paren {2 n + 1} \pi \\ 0 & : \paren {2 n + 1} \pi \le t \le \paren {2 n + 2} \pi \end {cases}$ for all integers $n$. === Graph of Half Wave Rectified Sine Curve === {{:Definition:Half Wave Rectified Sine Curve/Graph}}
Definition:Convolution Integral
Let $f$ and $g$ be real functions which are integrable. The '''convolution integral''' of $f$ and $g$ is defined as: :$\displaystyle \map f t * \map g t := \int_{-\infty}^\infty \map f u \map g {t - u} \rd u$
Definition:Convergent Sequence/Real Numbers/Graphical Illustration
The following diagram illustrates the first few terms of a convergent real sequence. :510px For the value of  $\epsilon$ given, a suitable value of $N$ such that: :$n > N \implies \size {x_n - l} < \epsilon$ is $6$.
Definition:Unbounded Divergent Sequence/Real Sequence/Positive Infinity
$\sequence {x_n}$ '''diverges to $+\infty$''' : :$\forall H \in \R_{>0}: \exists N: \forall n > N: x_n > H$ That is, whatever positive real number $H$ you choose, for sufficiently large $n$, $x_n$ will exceed $H$. We write: :$x_n \to +\infty$ as $n \to \infty$ or: :$\displaystyle \lim_{n \mathop \to \infty} x_n \to +\infty$
Definition:Unbounded Divergent Sequence/Real Sequence/Negative Infinity
$\sequence {x_n}$ '''diverges to $-\infty$''' : :$\forall H \in \R_{>0}: \exists N: \forall n > N: x_n < -H$ That is, whatever positive real number $H$ you choose, for sufficiently large $n$, $x_n$ will be less than $-H$. We write: :$x_n \to -\infty$ as $n \to \infty$ or: :$\displaystyle \lim_{n \mathop \to \infty} x_n \to -\infty$
Definition:Unbounded Divergent Sequence/Real Sequence/Infinity
$\sequence {x_n}$ '''diverges to $\infty$''' : :$\forall H > 0: \exists N: \forall n > N: \size {x_n} > H$
Definition:Local Basis/Local Basis for Open Sets
A '''local basis''' at $x$ is a set $\BB$ of open neighborhoods of $x$ such that: :$\forall U \in \tau: x \in U \implies \exists H \in \BB: H \subseteq U$ That is, such that every open neighborhood of $x$ also contains some set in $\BB$.
Definition:Local Basis/Neighborhood Basis of Open Sets
A '''local basis''' at $x$ is a set $\BB$ of open neighborhoods of $x$ such that every neighborhood of $x$ contains a set in $\BB$. That is, a '''local basis''' at $x$ is a neighborhood basis of $x$ consisting of open sets.