Proof

Suppose that there are only finitely many primes, say p₁, ..., p_r. Construct the integer m = p₁ ··· p_r + 1. Then m > 1. The integer m is not divisible by any of the p_i (i = 1, ..., r). The smallest divisor larger than 1 of m is a prime. This is a prime that is not in our list. Contradiction.