Cauchy criterion for series ($\forall \epsilon >0\exists N\forall m>n\ge N:|{\sum }_{k=n+1}^m{a}_k|<\epsilon$)

A series over the real numbers or complex numbers converges if and only if its tails satisfy: for every $\epsilon >0$, some $N$ makes $|{\sum }_{k=n+1}^m{a}_k|<\epsilon$ whenever $m>n\ge N$. This says exactly that its partial sums form a Cauchy sequence; sufficiency uses completeness.