If $H\subseteq G$ is a cocompact subgroup, then $G=KH$ for $K\subseteq G$ compact subgroup

Question

The question is in the title. $H\subseteq G$ is called cocompact if $G/H$ is a compact space.

This is a well-known fact but I couldn't prove it myself.

I would be happy as well for a reference, which I failed finding myself.

Thanks!

Edit: Mistake in the title, I meant $K$ compact subset in $G$. As already answered here, it is not possible if one requires $K$ sub-group.

Answer 1

There is no reference and this isn't wellknown, because it's easily seen to be false. Simple example: $G=\mathbb{Z}$ and $0<H<G$. $G/H$ is finite and therefore compact, but the only compact subgroup of $G$ is $0$ itself so that $K+H\neq G$. Similarly one could choose $G=\mathbb{R}$ and $0<H<G$ or $G=\mathbb{R}^n$ and any lattice of maximal dimension. And there are many, many more examples.