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

We set ${\displaystyle {}x_{n}:=\sum _{k=0}^{n}a_{k}}$. 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 ${\displaystyle {}\epsilon >0}$ there exists some ${\displaystyle {}n_{0}}$ such that for all

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

the estimate

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

holds. In the case of a series this just means

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