Cauchy's Lemma (Group Theory)/Proof 2

Proof
By First Sylow Theorem, $G$ has subgroups of order $p^r$ for all $r$ such that $p^r \divides \order G$.

Thus $G$ has at least one subgroup $H$ of order $p$.

As a Prime Group is Cyclic, $H$ is a cyclic group.

Thus by definition $H$ has an element of order $p$.

Hence the result.