Cauchy sequence

A sequenceMathworldPlanetmath x0,x1,x2, in a metric space (X,d) is a Cauchy sequenceMathworldPlanetmathPlanetmath if, for every real number ϵ>0, there exists a natural numberMathworldPlanetmath N such that d(xn,xm)<ϵ whenever n,m>N.

Likewise, a sequence v0,v1,v2, in a topological vector spaceMathworldPlanetmath V is a Cauchy sequence if and only if for every neighborhood U of 𝟎, there exists a natural number N such that vn-vmU for all n,m>N. These two definitions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath when the topologyMathworldPlanetmathPlanetmath of V is induced by a metric.

Title Cauchy sequence
Canonical name CauchySequence
Date of creation 2013-03-22 11:55:04
Last modified on 2013-03-22 11:55:04
Owner djao (24)
Last modified by djao (24)
Numerical id 10
Author djao (24)
Entry type Definition
Classification msc 54E35
Classification msc 26A03
Synonym fundamental sequence
Related topic MetricSpace