Real numbers/Series/Cauchy-criterion/Fact/Proof

Proof

We set ${}x_{n}:=\sum _{k=0}^{n}a_{k}$. The convergence of the series means the convergence of this sequence of partial sums. A real sequence converges if and only if it is a Cauchy-sequence. This means that for every ${}\epsilon >0$ there exists some ${}n_{0}$ such that for all

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

the estimate

${}\vert {x_{n}-x_{m}}\vert \leq \epsilon \,$

holds. In the case of a series this just means

${}\vert {x_{n}-x_{m}}\vert =\vert {\sum _{k=0}^{n}a_{k}-\sum _{k=0}^{m}a_{k}}\vert =\vert {\sum _{k=m+1}^{n}a_{k}}\vert \leq \epsilon \,.$

To fact