Let's see if I still remember second year math.

Suppose the number of primes is finite, call them p1, p2, ..., pn. Let P be their product, P = p1...pn. Since P > pi for all i, P+1 is not prime. Therefore some pk divides P+1. but pk also divides P since it is in the list of primes multiplied. Sp pk|((P+1)-P), i.e. pk|1, a contradiction. Therefore the number of primes is infinite.

That proof depends on the fundamental theorem of arithmetic, and a theorem that says that if p|X and p|Y then p|(X-Y).