PlanetMath: convergent sequence

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convergent sequence

(Definition)

A sequence $x_0, x_1, x_2, \dots$ in a metric space 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 such that $d(x,x_n) < \epsilon$ for all .

The point , 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 .

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

"convergent sequence" is owned by djao.

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Also defines:

limit point, limit, converge, diverge, divergent sequence

Keywords:

converge, diverge

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Cross-references: divergent, natural number, real number, point, metric space, sequence
There are 355 references to this entry.

This is version 5 of convergent sequence, born on 2001-10-27, modified 2005-06-01.
Object id is 601, canonical name is ConvergentSequence.
Accessed 23109 times total.

Classification:

AMS MSC:	40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
	54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability)

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