Reductio ad absurdum (Proof by negation):
Assume there are N primes.
Let the prime numbers be P1, P2, P3, … PN where PN is the largest prime number.
Construct the number P1 * P2 * P3 * … * PN + 1.
This number is not divisible by P1, P2 … PN, since there is always a remainder of 1.
Therefore no matter how high a prime we find, we can construct a number larger (much larger) than that one.
Incidentally, this also gives us a way to construct prime numbers.