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# Cauchy criterion for convergence

A series $\sum_{{i=0}}^{\infty}a_{i}$ in a Banach space $(V,\|\cdot\|)$ is convergent 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 convergent iff

$\|s_{{n+p}}-s_{n}\|=\|a_{{n+1}}+a_{{n+2}}+\cdots+a_{{n+p}}\|<\varepsilon.$ |

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

40A05*no label found*

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## Corrections

the the by yark ✓

use \cdots by Mathprof ✓

contains own proof by rspuzio ✓

Not quite by rm50 ✓

Contains own proof by rm50 ✓

use \cdots by Mathprof ✓

contains own proof by rspuzio ✓

Not quite by rm50 ✓

Contains own proof by rm50 ✓

## Versions

(v14) by mathwizard 2013-03-22

## Comments

## Unproven

The entry appears as unproven although it is not.