I'm not looking for the solution, but a helpful first step.

So far, I realize that if $\displaystyle H \cap K = \{e\}$, then $\displaystyle [G : H \cap K $] is infinite. More generally, if $\displaystyle o(H \cap K) < \infty$, then $\displaystyle [G : H \cap K]$ is infinite.

Problem is, I don't see how to show anything more than $\displaystyle e \in H \cap K $

Thanks for your help.