We will prove that there are an infinite number of prime numbers.

Let's presume the contrary:
There are only a finite number of prime numbers: p₁, p₂ ... p_n. (A)

Then let m be:
m = p₁p₂ ... p_n+1

Let d be the smallest divisor of m which is different than 1. Any divisor of d is also a divisor of m so d hasn't any divisors that are smaller then d and are different the d. So d is prime.

Then d is one of the p_i, with i from 1 to n. This is not possible because d is a divisor of m but p_i is not a divisor of m. So we have a contradiction it means that (A) is false.

In conclusion:
There are an infinite number of prime numbers.