Definition:Convergent Sequence

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Definition

Topological Space

Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $A$.

Then $\sequence {x_n}$ converges to the limit $\alpha \in S$ if and only if:

$\forall U \in \tau: \alpha \in U \implies \paren {\exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in U}$


Metric Space

See Convergent Sequence in Metric Space for equivalent definitions of a convergent sequence in a Metric Space


Let $M = \struct {A, d}$ be a metric space or a pseudometric space.

Let $\sequence {x_k}$ be a sequence in $A$.

$\sequence {x_k}$ converges to the limit $l \in A$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \map d {x_n, l} < \epsilon$


Normed Division Ring

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

Let $\sequence {x_n} $ be a sequence in $R$.


The sequence $\sequence {x_n}$ converges to the limit $x \in R$ in the norm $\norm {\, \cdot \,}$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x} < \epsilon$


Normed Vector Space

Let $\struct {X, \norm {\,\cdot \,} }$ be a normed vector space.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.

Let $L \in X$.


The sequence $\sequence {x_n}_{n \mathop \in \N}$ converges to the limit $L \in X$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - L} < \epsilon$


Test Function Space

Let $\map \DD {\R^d}$ be the test function space with the compact support $K \subseteq \R^d$.

Let $\sequence {\phi_n}_{n \mathop \in \N}$ be a sequence in $\map \DD {\R^d}$.

Let $\phi \in \map \DD {\R^d}$ be a test function.

Let $D^k := \dfrac {\partial^{k_1 + k_2 + \ldots + k_d}} {\partial x_1^{k_1} \partial x_2^{k_2} \ldots \partial x_d^{k_d} }$ be a partial differential operator with the multiindex $k = \tuple {k_1, k_2, \ldots, k_d}$.

Suppose:

$\forall n \in \N : \forall x \in \R^d \setminus K : \map {\phi_n} x = 0$

Suppose $\sequence {\phi_n}_{n \mathop \in \N}$ converges uniformly to $\phi$.

Suppose that for every multiindex $k$ the sequence $\sequence {D^k \phi_n}_{n \mathop \in \N}$ converges uniformly to $D^k \phi$.


Then the sequence $\sequence {\phi_n}_{n \mathop \in \N}$ converges to $\phi$ in $\map \DD {\R^d}$.


This can be denoted:

$\phi_n \stackrel \DD {\longrightarrow} \phi$


Schwartz Space

Let $\map \SS \R$ be the Schwartz space.

Let $\sequence {\phi_n}_{n \mathop \in \N}$ be a sequence in $\map \SS \R$.

Let $\phi \in \map \SS \R$ be a Schwartz test function.

Suppose:

$\ds \forall l, m \in \N : \lim_{n \mathop \to \infty} \sup_{x \mathop \in \R} \size {x^l \map {\phi_n^{\paren m} } x} = 0$

where:

$\phi^{\paren m}$ denotes the $m$th derivative of $\phi$
$\sup$ denotes the supremum.


Then the sequence $\sequence {\phi_n}_{n \mathop \in \N}$ converges to $\mathbf 0$ in $\map \SS \R$.

This can be denoted:

$\phi_n \stackrel \SS {\longrightarrow} \mathbf 0$


Standard Number Fields

When $A$ is one of the standard number fields $\Q, \R, \C$, and the metric $d$ is the usual (Euclidean) metric, the condition on convergence becomes:


Real Numbers

Let $\sequence {x_k}$ be a sequence in $\R$.

The sequence $\sequence {x_k}$ converges to the limit $l \in \R$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {x_n - l} < \epsilon$

where $\size x$ denotes the absolute value of $x$.


Rational Numbers

Let $\sequence {x_k}$ be a sequence in $\Q$.

$\sequence {x_k}$ converges to the limit $l \in \R$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {x_n - l} < \epsilon$

where $\size x$ is the absolute value of $x$.


Complex Numbers

Let $\sequence {z_k}$ be a sequence in $\C$.


$\sequence {z_k}$ converges to the limit $c \in \C$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$

where $\cmod z$ denotes the modulus of $z$.


$p$-adic Numbers

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\sequence {x_n} $ be a sequence in $\Q_p$.


The sequence $\sequence {x_n}$ converges to the limit $x \in \Q_p$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x}_p < \epsilon$


Also known as

A convergent sequence is also known by its variant converging sequence.

In the interests of consistency, the latter term is avoided on $\mathsf{Pr} \infty \mathsf{fWiki}$.


Also see

  • Results about convergent sequences can be found here.


Sources

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