Normal p-Subgroup contained in All Sylow p-Subgroups

Theorem
Let $G$ be a finite group.

Let $p$ be a prime number.

Let $H$ be a normal subgroup of $G$ which is a $p$-group.

Then $H$ is a subset of every Sylow $p$-subgroup of $G$.

Proof
By Finite Group is p-Group iff Order is Power of p:
 * $\order H = p^k$

for some $k \in \Z_{>0}$.