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# convergent sequence

A sequence $x_{0},x_{1},x_{2},\dots$ in a metric space $(X,d)$ is a *convergent sequence* if there exists a point $x\in X$ such that, for every real number $\epsilon>0$, there exists a natural number $N$ such that $d(x,x_{n})<\epsilon$ for all $n>N$.

The point $x$, if it exists, is unique, and is called the *limit point* or *limit* of the sequence. One can also say that the sequence $x_{0},x_{1},x_{2},\dots$ *converges* to $x$.

A sequence is said to be *divergent* if it does not converge.

Defines:

limit point, limit, converge, diverge, divergent sequence

Keywords:

converge, diverge

Related:

AxiomOfAnalysis, BolzanoWeierstrassTheorem, Sequence

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

54E35*no label found*40A05

*no label found*

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