40A05-CauchyCriterionForConvergence.tex

\documentclass[12pt]{article}
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\pmcanonicalname{CauchyCriterionForConvergence}
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\pmtitle{Cauchy criterion for convergence}
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\pmtype{Theorem}
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A series $\sum_{i=0}^\infty a_i$ in a Banach space $(V,\|\cdot\|)$ is \PMlinkid{convergent}{2311} 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 $p\geq1$.


\subsection*{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 \PMlinkescapetext{convergent} iff
$$\|s_{n+p}-s_n\|=\|a_{n+1}+a_{n+2}+\cdots+a_{n+p}\|<\varepsilon.$$
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