**Theorem 1: There are an infinite number of primes**

(1) Assume that $p_n$ is the largest prime.

(2) Let $q = p_n! + 1$

(3) Since $q > p_n$, $q$ cannot be a prime (by our assumption).

(4) So, therefore a prime $p$ divides $q$

(5) But if $p \le p_n$, then $p | p_n!$ and $p | (p_n! + 1) - p_n! = 1$ which is impossible.

(6) Therefore, we have a contradiction since $p > p_n$ but $p$ is a prime.

QED