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?

In the proof that is attributed to Euclid, what you are really doing is this: For any integer $n \geq 1$ you are producing a number, namely one plus the product of all primes less than or equal to $n$, whose prime factors have to be greater than the biggest prime less than $n$.
Therefore, given the fundamental theorem of arithmetic, Euclid's proof can be modified in such a way that it actually proves that for any integer $n \geq 1$ there exists a prime $p \geq n$, that is, there are arbitrarily large primes.
