I am having difficulties in solving the following problem.
Let $G$ be a group with a free subgroup of rank $2$. Let $H\leq G$ be such that $[G:H]<\infty$. Then $H$ also contains a free subgroup of rank $2$.
We know by Nielsen-Schreier theorem that a subgroup of a free group is also free. But in this problem $G$ is not necessarily free but contains a free subgroup. How to approach this problem? Any hint or idea will be highly appreciated. Thanks in anticipation.