Why $U$ generates $G$ as Lie group?

Question

In line 2 of the proof, why is their intersection non-empty?

Answer 1

$U$ is a an open set containing $e$. Thus $gU$ is open and contains $g$ (note that multiplication by an element of a topological group is a homeomorphism). But $g \in \partial H$ so every neighborhood of $g$ intersects $H$ non-trivially