convergent sequence

A sequence $x_0,x_1,x_2,\mathrm{\dots }$ in a metric space $(X,d)$ is a convergent sequence if there exists a point $x\in X$ such that, for every real number $ϵ>0$, there exists a natural number $N$ such that 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 $x_0,x_1,x_2,\mathrm{\dots }$ converges to $x$.

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

Title	convergent sequence
Canonical name	ConvergentSequence
Date of creation	2013-03-22 11:55:07
Last modified on	2013-03-22 11:55:07
Owner	djao (24)
Last modified by	djao (24)
Numerical id	10
Author	djao (24)
Entry type	Definition
Classification	msc 54E35
Classification	msc 40A05
Related topic	AxiomOfAnalysis
Related topic	BolzanoWeierstrassTheorem
Related topic	Sequence
Defines	limit point
Defines	limit
Defines	converge
Defines	diverge
Defines	divergent sequence