I am trying to prove first Sylow Theorem using the Lemma: if $G$ is a finite group such that has a Sylow $p$-subgroup and $H\subset G$, then $H$ has a Sylow $p$-subgroup.
The way I want to go about proving first Sylow Theorem using lemma is to notice that any finite group $G$ injects into $S_{|G|}$.
Given the Lemma, it suffices to show that $S_{|G|}$ has a Sylow $p$-subgroup. If $p^k$ is the highest power of $p$ such that $p^k| |G|$, then I can easily see that $\langle(123\cdots p^k)\rangle\subset S_{|G|}$ is a subgroup with order $p^k$. But this doesn't finish the proof as $|G|!$ may be divisible by $p^{k+1}$, which then means this subgroup is not a Sylow $p$-subgroup of $S_{|G|}$.
Would there be hints about what steps I should make in this way of proving the first Sylow Theorem?