This was a question posed to some students but I left without a proper answer. Here's the proof.

Proof by Contradiction
Let the set of prime numbers be {p1, p2, p3 ..., pn} where n is finite.
We can write a number x as x = p1 * p2 * p3 ... * pn + 1
Such a number x, when divided by any prime p gives a remainder of 1, hence it is divisible by 1 and itself only which mean x is a prime too and the first statement must be wrong.

Hence the set of prime number must be infinite in size

Simple?