Cauchy criterion for convergence

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

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

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 $\epsilon >0$ 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