Real numbers/Series/Cauchy-criterion/Fact

Let

$\sum _{k=0}^{\infty }a_{k}$

be a series of real numbers.

Then the series is

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

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

the estimate

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

holds.

Proof, Write another proof