Assume p1, ..., pn is a finite list of primes. The sum p1+...+pn+1 is divisible by a prime, because every natural number> 1 is. However, it's not divisible by p1,...,pn, hence there must be an additional prime not in the list.
(I think you're right though that GP's "contradiction" doesn't work)
Never thought of using "by definition, all numbers can be divided by a prime" tu merge the two cases. It's not that shorter, but is IMHO quite elegant, I'll remember it. Thanks for correcting me.
Well, it's not by definition, but "every number is divisible by a prime" is fairly obvious (just keep dividing until you reach a prime) and can technically be proven by using (strong) induction.
Assume p1, ..., pn is a finite list of primes. The sum p1+...+pn+1 is divisible by a prime, because every natural number> 1 is. However, it's not divisible by p1,...,pn, hence there must be an additional prime not in the list.
(I think you're right though that GP's "contradiction" doesn't work)