Here’s a quick neat one.

Note that any two consecutive integers n and n+1 have no factors in common (n>1). So the number

n (n+1)

must be divisible by at least two different prime numbers. Similarly, if we add one to this new number than we get the number

n (n+1) + 1

which must contain a new prime factor. Therefore the number

(n (n+1) ) (n (n+1) + 1)

must be divisible by at least three different prime numbers. Carrying on this process creates numbers with as many prime factors as you like. Therefore, the number of primes is unbounded.