Absolute convergence implies Cauchy

Absolute convergence implies Cauchy: Let $\sum_{n=1}^\infty a_n$ be a series of real or complex numbers, and let $s_n=\sum_{k=1}^n a_k$ be its partial sums . If

converges, then $(s_n)$ is a Cauchy sequence; equivalently, for every $\varepsilon>0$ there exists $N$ such that for all integers $m>n\ge N$,

This is the series form of the Cauchy criterion and is a standard route to absolute convergence implies convergence .