How to prove the quotient group is compact?

Assume $G$ is an abelian topological group. Let $Bsubset A$ be its two subgroups, $A$ closed and $B$ discrete. $C$ is a compact subset in $G$. Suppose $Asubset BC$, can we prove $A/B$ is also compact?

I want to show $A/B$ can be identified with a closed subset of $C$, but I can not give a strict proof. Could you help me prove it or give a counterexample?