Use Sylow’s Theorem to prove Cauchy’s Theorem

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.1
Solution: Let $G$ be a finite group and $p$ a prime dividing $|G|$. By Sylow’s Theorem, $G$ has a Sylow $p$-subgroup $P$. Since $P$ is a $p$-group, $Z(P)$ is nontrivial. Now $Z(P)$ is an abelian $p$-group, so that (by Cauchy’s Theorem for abelian groups) there exists an element $x \in Z(P) \leq P \leq G$ of order $p$.