Proof? How's this?
Given any finite collection of primes, p_1 ,...,p_n, form the number N=1+p_1 p_2...p_n. Then N has a prime factor, but none of p_1, ...,p_n is such (since they all give a remainder of 1 when divided into N). Hence, there is a prime not in the list.
Conclusion: there are an infinite number of prime integers.
Given any finite collection of primes, p_1 ,...,p_n, form the number N=1+p_1 p_2...p_n. Then N has a prime factor, but none of p_1, ...,p_n is such (since they all give a remainder of 1 when divided into N). Hence, there is a prime not in the list.
Conclusion: there are an infinite number of prime integers.