Let $G$ is a group and $H<G$ such that $|G-H|<\infty$. Prove that $|G|<\infty$.
Truthfully, there is a hint for it:
$H$ cannot be an infinite subgroup.
It is clear if $|H|<\infty$, since $|G-H|<\infty$ then $|G|<\infty$ and problem will be solved. But cannot understand why "$H$ cannot be an infinite subgroup". Thanks for your help.