Real series/Absolute convergence and convergence/Fact/Proof

Proof

Let ${}\epsilon >0$ be given. We use the Cauchy-criterion. Since the series converges absolutely, there exists some ${}n_{0}$ such that for all ${}n\geq m\geq n_{0}$ the estimate

${}\vert {\sum _{k=m}^{n}\vert {a_{k}}\vert \,}\vert =\sum _{k=m}^{n}\vert {a_{k}}\vert \leq \epsilon \,$

holds. Therefore,

${}\vert {\sum _{k=m}^{n}a_{k}}\vert \leq \vert {\sum _{k=m}^{n}\vert {a_{k}}\vert \,}\vert \leq \epsilon \,,$

which means the convergence.

To fact