First Sylow Theorem/Corollary/Proof 1

Proof
Let $\order G = k p^r$ where $p \nmid k$.

From the First Sylow Theorem, $G$ has a subgroup $S$ of order $p^r$.

From (need to find it), $S$ itself has subgroups of order $p^n$ for all $n \in \set {1, 2, \ldots, r}$.