Proposition 9

Let $H$ be a dense subgroup of a topological group $G$. If $H$ is generated by every neighbourhood of the identity element in $H$, then $G$ is generated by every neighbourhood of the identity element in $G$.

Let $V$ be any symmetric neighbourhood of $e$ in $G$. Then $V \cap H$ is a neighbourhood of $e$ in $H$, and hence generates $H$. It follows that $V$ generates a subgroup $H'$ which contains $H$; but $H'$ is open and closed (no. 1, Corollary to Proposition 4), and therefore contains $\overline{H} = G$.