Cauchy criterion for convergence

A series i=0ai in a Banach space (V,) is iff for every ε>0 there is a number N such that


holds for all n>N and p1.


First define


Now, since V is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, (sn) convergesPlanetmathPlanetmath if and only if it is a Cauchy sequenceMathworldPlanetmathPlanetmath, so if for every ε>0 there is a number N, such that for all n,m>N holds:


We can assume m>n and thus set m=n+p. The series is iff

Title Cauchy criterion for convergence
Canonical name CauchyCriterionForConvergence
Date of creation 2013-03-22 13:22:03
Last modified on 2013-03-22 13:22:03
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 14
Author mathwizard (128)
Entry type Theorem
Classification msc 40A05