Proving that $H_e$ is a subgroup of $G$

Question

I'm having some trouble understanding the proof for the following proposition:

Let $G$ be a topological group, $H$ be a normal subgroup and $H_e$ be the connected component of $e$ with respect to $H$. Then $H_e$ is a normal subgroup of $G$.

They started by proving that, for any $g\in G, gH_eg^{-1}\subseteq H_e$, but then the proof ended there. The author didn't prove that $H_e$ is a group in the first place, maybe because it's trivial, but either way I can't figure out how to prove this.

How can this be done?

Answer 1

Since I cannot make a comment, I’ll give you a hint as an answer:

Continuous images of connected sets are connected and the group multiplication and inversion are continuous maps.