How to prove that there are infinitely many primes without using contradiction 
How can I prove that there are infinitely many primes without using contradiction?

I know the proof that is (not) by Euclid saying there are infinitely many primes. It assumes that there is a finite set of primes and then obtains one that is not in that set. 
Say for some reason I want to prove it without using contradiction. Maybe I'm contradictaphobic. How could I go about doing this?

Let $F_n=2^{2^n}+1$ be the sequence of Fermat numbers. It is relatively easy to show that, for distinct $i,j$ we have $\gcd(F_i,F_j)=1$. Therefore, if we let $p_n$ be a sequence of primes such that $p_n$ divides $F_n$ (which is possible, since every number has at least one prime divisor), then each $p_n$ is distinct (since $p_n>1=\gcd(F_i,F_j)$), and there are infinitely many of them.
