Real numbers/Series/Cauchy-criterion/Fact

convergent if and only if the following Cauchy-criterion holds: For every ${\displaystyle {}\epsilon >0}$ there exists some ${\displaystyle {}n_{0}}$ such that for all

${\displaystyle {}n\geq m\geq n_{0}\,}$

the estimate

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