Exercise 5, Appendix 0

a) Let $G$ be a complete group such that every neighbourhood of the identity element $e$ in $G$ contains an open subgroup of $G$. Show that a sequence $(x_n)$ of points of $G$ is multipliable if and only if $\lim_{n \to \infty} x_n = e$.

b) Let $G$ be a complete group in which every neighbourhood of $e$ contains a normal open subgroup of $G$. Show that a family $(x_i)_{i \in I}$ of points of $G$ is multipliable if and only if $\lim x_i = e$ with respect to the filter of complements of finite subsets of $I$ (use Exercise 3).