Euclid's Theorem/Corollary 1/Proof 1

Proof
Assume that there are only finitely many prime numbers, and that there is a grand total of $n$ primes.

Then it is possible to define the set of all primes:
 * $\mathbb P = \set {p_1, p_2, \ldots, p_n}$

From Euclid's Theorem, however, we can always create a prime which is not in $\mathbb P$.

So we can never create a finite list of all the primes, because we can guarantee to construct a number which has prime factors that are not in this list.

Thus, there are infinitely many prime numbers.