Exercise 1, § 5

Let $(x_i)_{i \in I}$ be a summable family in a complete Hausdorff commutative group $G$, and let $\mathscr{S}$ be a set of subsets of $I$, directed with respect to the relation $\subset$ and forming a covering of $I$. Show that $\sum_{i \in I} x_i = \lim_{\mathscr{S}} \sum_{j \in J} x_i$, where the limit is taken with respect to the section filter of $\mathscr{S}$.