Let $G$ be a locally compact, connected topological group. Show that $G$ is a paracompact.
I was trying to prove it with defining $U_n$ by defining
$U_1$ to be a symmetric neighborhood of $e$ having compact closure, and $U_{n+1} =\bar U_n \circ U_1$.
Now I have only step to show $U=\cup U_n$ is subgroup of $G$. However, I was stuck to show
For each $a\in U$, $a^{-1} \in U$