# Cauchy sequence

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

Likewise, a sequence $v_{0},v_{1},v_{2},\dots$ in a topological vector space $V$ is a Cauchy sequence if and only if for every neighborhood $U$ of $\mathbf{0}$, there exists a natural number $N$ such that $v_{n}-v_{m}\in U$ for all $n,m>N$. These two definitions are equivalent when the topology of $V$ is induced by a metric.

Title Cauchy sequence CauchySequence 2013-03-22 11:55:04 2013-03-22 11:55:04 djao (24) djao (24) 10 djao (24) Definition msc 54E35 msc 26A03 fundamental sequence MetricSpace