extension of discrete groups

Question

Suppose $A$ and $C$ are discrete groups, $B$ is an extension of $C$ by $A$, i.e. we have a short exact sequence:

$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$ Is $B$ also discrete?

Answer 1

Yes.

Consider a convergent sequence $\{b_n\}_n$ in $B$. Then its image in $C$ converges, hence is eventually constant. Then for some $N\gg 0$, $b_nb_N^{-1}$ is in $A$ for all $n\gg 0$, where it still converges, hence is eventually constant.