Euclid's Theorem/Corollary 1/Proof 2

Corollary to Euclid's Theorem
There are infinitely many prime numbers.

Proof
Assume that there are only finitely many prime numbers.

Let $p$ be the largest of these.

Then from Existence of Prime between Prime and Factorial there exists a prime number $q$ such that:
 * $p < q \le p! + 1$

So there cannot be such a $p$.