A topological group question about generators

Question

If $G$ is connected topological group and $e \in V$, $V$ is open. Then prove that $V$ is a set of generators for $G$.

Answer 1

Hint: define $V^2:=\{v_1v_2,v_1,v_2\in V\}$ and similarly, $V^n:=\{v_1\dots v_n,v_i\in V, i\in [n]\}$. Let $H:=\bigcup_{n\geqslant 1}V^n$: then $H$ is an open subgroup and it is generated by $V$ (we can assume that $V^{-1}:=\{v^{-1},v\in V\}$ is contained in $V$).