Exercise 2, § 5

Let $G$ be a complete Hausdorff commutative group such that every neighbourhood of $o$ in $G$ contains an open subgroup of $G$ (this is the case in particular if $G$ is locally compact and totally disconnected; cf. § 4, Exercise 18). A family $(x_i)_{i \in I}$ of points of $G$ is summable if and only if $\lim x_i = o$ with respect to the filter of complements of finite subsets of $I$. Hence show that every convergent series in $G$ is commutatively convergent.