[bluecheese33]

It is a common simplification of Euclid's proof to convert it to a proof by contradiction. In fact, I believe I was taught this version of the proof in an early undergraduate math class. This version is not constructive, actually the first example of the wikipedia article on constructive proofs points this out:

Euclid's proof is constructive. But a common way of simplifying Euclid's proof postulates that, contrary to the assertion in the theorem, there are only a finite number of them, in which case there is a largest one, denoted n. Then consider the number n! + 1 (1 + the product of the first n numbers). Either this number is prime, or all of its prime factors are greater than n. Without establishing a specific prime number, this proves that one exists that is greater than n, contrary to the original postulate.