What is the relation between syndetic subgroup and finitely index subgroup?

Question

Let $G$ be a topological group. Subgroup $H$ of $G$ is called syndetic if there is compact set $K\subseteq G$ such that $G=KH$.

Gottschalk and Hedlund ( in Topological Dynamics, P. 12) said that if $G$ is discrete and $H$ is a subgroup of $G$, then $H$ is syndetic if and only if $H$ is finite index in $G$. I can not prove it.

Can some body help me to know proof it? Thanks

Answer 1

Hint: If $K$ is a compact set in a discrete group then $K$ is finite.