Prove that there are an infinite number of prime numbers.

To begin, we assume that there are only a finite number of prime numbers, say $p_1,p_2,p_3,\dots ,p_{n-1},p_n$.

Consider the number $q=p_1\times p_2\times p_3\times \dots \times p_{n-1}\times p_n+1$

The number $q$ is not divisible by any of the prime numbers $p_i$, since it leaves a remainder of 1 when it is divided by any of them. So, $q$ must be divisible by some prime that is not in the list, by the Fundamental Theorem of Arithmetic. This is a contradiction to our assumption of having the complete list of prime numbers. Hence, there must be an infinite number of prime numbers.