The following proof is from Introduction to Mathematics (Devlin). I am not trying to disprove Devlin or Euclid. I’m trying to understand if the questions my mind asks are unwarranted and why.
Statement: There are infinitely many prime numbers.
(Proof: First part) "The truth of this statement can be proved by an ingenious argument known to Euclid.1 The idea is to show that if we list the primes in increasing order as $p_1, p_2, p_3,..., p_n,...$ then the list must continue forever. (The first few members of the sequence are: $p_1 = 2, p_2 = 3, p_3 = 5, p_4 = 7, p_5 = 11$, and so on.) Consider the list up to some stage $n$: $p_1, p_2, p_3,..., p_n$ The goal is to show that there is another prime that can be added to the list. Provided we do this without assigning n a specific value, this will imply at once that the list is infinite.
Let $N$ be the number we get when we multiply together all the primes we have listed so far and then add $1$, i.e., $N = (p_1p_2\cdots p_n) + 1$ Obviously, $N$ is bigger than all the primes in our list, so if $N$ is prime, we know there is a prime bigger than $p_n$, and hence the list can be continued. (We are not saying that $N$ is that next prime. In fact, $N$ will be much bigger than $p_n$, so it is unlikely to be the next prime.)"
Q1) Why is the proof legit? This proof employs only on $N=(1…5)+1$. How, without calculating an answer for EVERY $p_X...p_N$, can one assume the proof will continue to be true?
Q2) How about if $N=(p1…p_\infty)+1$?
Q3) How can it be said “hence the list can be continued”?