CauchyCriterionForConvergence 2003-01-16 05:12:35 2007-12-15 03:32:22 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 $$\|s_{n+p}-s_n\|=\|a_{n+1}+a_{n+2}+\cdots+a_{n+p}\|<\varepsilon.$$