Exercise 31, § 2

Let $G$ be a connected topological group, $H$ a subgroup of $G$, $U$ an open subset of $G$ such that $H = UH$. Let $V = H \cap (U^{-1}U)$; show that $H = V^\infty$ [show that if $A = H \cap C(V^\infty)$, then the sets $U.V^\infty$ and $U.A$ do not intersect].