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?

The simplest direct proof I know is one which involves infinite series. One can show that $\sum\limits_{p\text{ prime}}\frac{1}{p}$ diverges. Particularly, it means that there cannot be finitely many primes since the sum of finitely many nonzero numbers is finite. (There is a very slight contradiction - or contrapositive - argument in here if you look closely enough.) That said, you should definitely combat your contradictaphobia... proofs by contradiction are everywhere in mathematics. Hamstringing yourself by not accepting or feeling at ease with such proofs will make mathematics quite challenging at times.
