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 $2$.

Let $F$ be the free subgroup of rank $2$ in $G$.

Then $|G:H|$ finite implies that $k := |F:H \cap F|$ is also finite, and by the Nielsen-Schreier Theorem $H \cap F$ is free of rank $k+1$.

So $H \cap F$ and hence also $H$ contains a free subgroup of rank $2$.

Let $n=|G:H|$ be the index of $H$ in $G$, and let $a, b\in G$ be the generators of $F$. Then $\langle a^{n!}, b^{n!}\rangle$ is free of rank two, and is contained in $H$.