Let $H$ and $K$ be subgroups of a group $G$. Then is the following true ?
If $[G : H] < ∞$ and $[G : K] < ∞$, then $[G : H ∩ K] < ∞$.
I think this's false because there's still a case that $H ∩ K$ could be empty. But the textbook requires to prove this statement. So I am a bit confusing on this point.. any help?