Finite p-groups have subgroups of all p-power orders

Proposition (Subgroups of all orders in a finite p-group). Let $p$ be a prime and let $G$ be a finite p-group with $|G|=p^n$ . Then for every integer $k$ with $0\le k\le n$ there exists a subgroup $H\le G$ such that $|H|=p^k$ .

Remarks

Context. This is a structural strengthening of Lagrange's theorem for $p$ -groups: not only do subgroup orders divide $|G|$ , all intermediate $p$ -powers actually occur. It is proved by induction using the existence of nontrivial center.