Theorem 1 (Cauchy's criterion)

In a Hausdorff commutative group $G$, in order that a family $(x_i)_{i \in I}$ should be summable it is necessary that, for each neighbourhood $V$ of the origin in $G$, there is a finite subset $J_0$ of $I$ such that $\sum_{i \in K} x_i \in V$ for all finite subsets $K$ of $I$ which do not meet $J_0$. This necessary condition is also sufficient if $G$ is complete.

Thus by taking away a (sufficiently large) finite number of terms from the family $(x_i)$, every finite partial sum of the surviving subfamily has to be as close to $o$ as we please.

An immediate consequence of the first part of Theorem 1 is the following proposition: