# 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.

 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