Apparently I misremembered the details of the proof. I thought that it started with (P1, P2, P3, ..., PN) being a list of all the primes from 2 up to PN. Since

Apparently I misremembered the details of the proof. I thought that it started with (P1, P2, P3, ..., PN) being a list of all the primes from 2 up to PN. Since (P1 * P2 * ... * PN) + 1 either is prime or has a prime factor not in the list, that prime factor must be greater than PN. Apparently (according to Wikipedia's summary) Euclid started with an arbitrary list of primes, and showed that for any such list there is a prime not in the list. Either method works to show that there are infinitely many primes.