There are many different proofs of this theorem.

Each depends upon what you have to work with.

If you know that there is a one-to-one correspondence between the prime integers and the positive integers. Suppose that is that mapping. Now if is a finite subset of then consider the function . Now if we use theunique factorizationtheorem then the function is one-to-one. Thus the set of all finite subset of maps injectively into . Thus it is countable.