Exercise 6, § 3

Show that the least upper bound of the right and left uniformities on a topological group $G$ (Chapter II, § 2, no. 5) is a uniformity compatible with the topology of $G$: it is called the two-sided uniformity on $G$. Show that every Hausdorff topological group is isomorphic to a dense subgroup of a topological group which is complete with respect to its two-sided uniformity.