Writing G as a product of groups

Question

$G$ is a connected, locally compact group satisfying the second axiom of countability, and $C$ is a discrete central subgroup of $G$ such that $G/C$ is compact. In the book I am reading (Varadarajan, Lie Groups, Lie Algebras, and their Representations, Lemma 4.11.1) it says that one can choose a compact set $D$ such that $G=C (\text{int} D)$. Why is this true?

Answer 1

Get yourself a compact neighbourhood $N=N^{-1}$ of the identity in $G$, and let $U=N^\circ$. Since $\bigcup_{n\geq 1}U^n$ is an open subgroup of $G$, and $G$ is connected, we have $\bigcup_{n\geq 1} U^n=G$. This means $CU^n=\{Cu_1\dots u_n\mid u_i\in U\forall i\}$, for $n\geq 1$, forms an open cover of $C\backslash G=G/C$ (since $C$ is central), which is compact. So there is a finite subcover. Now $(CU^n)_{n\geq 1}$ is nested, so there is some $n\in\mathbb{N}$ such that $CU^n=G/C$. Since $U\subseteq N$, we have $D=N^n$ is compact, $D^\circ\supseteq U^n$ and $C(D^\circ)=G$.