I am given $G$ locally compact group, and I want to show that there exists a clopen subgroup $H$ of $G$ that is $\sigma$-compact.
So here's what I did so far:
for $e \in U$, where $U$ is a nbhd of the identity element we know that $x \in x\bar{U}$, and $x\bar{U}$ is compact in $G$, now take some finite open cover of this set,$\cup_{i=1}^{n} V_i$, again this set is open nbhd of $x$ so again its closure is compact, and its closure equals the union of $\bar{V_i}$, now I am kind of stuck, I wish I could take this set as the clopen subgroup, but I don't think that I know that it's closed under multiplication, right?