Exercise 3, § 5

Let $(x_i)_{i \in I}$ be a family of points of a Hausdorff commutative group $G$. For every finite subset $H$ of $I$, let $s_H$ denote $\sum_{i \in H} x_i$, and let $A$ be the set of cluster points of the mapping $H \to s_H$ with respect to the directed set $\mathfrak{F}(I)$. For every finite subset $J$ of $I$, let $\Phi(J)$ denote the closure of the set of points $s_H$, where $H$ runs through the finite subsets of $I$ such that $H \cap J = \varnothing$. Show that if $A \neq \varnothing$ then the set $A - A$ (i.e. the set of all $x - x'$ where $x \in A$ and $x' \in A$) is the same as the set $B = \bigcap_{J \in \mathfrak{F}(I)} \Phi(J)$; and that in any case $B + B \subset B$. Deduce that if $A$ is not empty, then $B$ is a closed subgroup of $G$ and that $A$ is a coset of $B$ in $G$.