Proving Cauchy's theorem (group theory)

Here is an easy proof of this due to McKay. Consider the set $$ S = \{(a_1,\ldots,a_p) \in G^p : a_1\cdots a_p = e\}. $$ Also, define $A = \{x \in G : x^p = e\}$. Clearly $|S| = |G|^{p-1}$. Every non-constant vector in $S$ has $p$ shifts (including itself) also in $S$, and so $p \mid |S| - |A|$. Since $p \mid |G|$, we conclude that $p \mid |A|$, and in particular $|A| \geq p$ (since $e \in A$). Apart from $e$, all other elements in $A$ have order $p$.

Any element in your Sylow $p$-subgroup generates a cyclic group of $p$th-power order. An element of order $p$ in this cyclic group is an element of order $p$ in the Sylow subgroup.