# Cauchy criterion for convergence

A series $\sum_{i=0}^{\infty}a_{i}$ in a Banach space $(V,\|\cdot\|)$ is http://planetmath.org/node/2311convergent iff for every $\varepsilon>0$ there is a number $N\in\mathbb{N}$ such that

 $\|a_{n+1}+a_{n+2}+\cdots+a_{n+p}\|<\varepsilon$

holds for all $n>N$ and $p\geq 1$.

## Proof:

First define

 $s_{n}:=\sum_{i=0}^{n}a_{i}.$

Now, since $V$ is complete, $(s_{n})$ converges if and only if it is a Cauchy sequence, so if for every $\varepsilon>0$ there is a number $N$, such that for all $n,m>N$ holds:

 $\|s_{m}-s_{n}\|<\varepsilon.$

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

 $\|s_{n+p}-s_{n}\|=\|a_{n+1}+a_{n+2}+\cdots+a_{n+p}\|<\varepsilon.$
Title Cauchy criterion for convergence CauchyCriterionForConvergence 2013-03-22 13:22:03 2013-03-22 13:22:03 mathwizard (128) mathwizard (128) 14 mathwizard (128) Theorem msc 40A05