Let $G$ be a $p$-group, where $p$ is prime, and let $H$ be a proper subgroup of $G$. I want to show that $H$ is contained in a normal subgroup of index $p$ in $G$.
I know that subgroups of index $p$ are normal, so all I have to do is show that $H$ is contained in such a subgroup. I also know that the normalizer $N(H)$ of $H$ in $G$ strictly contains $H$, so I was thinking of looking at the sequence of normalizers $N(N(H)), N(N(N(H))), \dots$, and arguing that this sequence eventually gives me a subgroup of index $p$. I'm not sure how to begin making this argument though.
Any help is appreciated!