Let $G$ be a group that every element have infinite order. If $G$ has a cyclic subgroup with finite index then $G$ is cyclic.
First I show that $G$ is finitely generated. If $\langle x\rangle$ be the cyclic subgroup that is mentioned above then assuming that $|G:\langle x\rangle|=n$, $G$ has elements like $g_2,\ldots,g_n$ such that $G=\langle x,g_2,\ldots,g_n\rangle$.
Now I want to show that $G=\langle x\rangle$ but I don't know how to continue.