Suppose that there exists a finite amount of Primes and the list is as follows:
(p1, p2, p3.....pn).
n being the amount of primes.
Now we shall assume that the integer obtained by multiplying the primes is "N".
Therefore N = (p1 p2 p3.....pn).
N is obviously not Prime because it is divisible by all the other primes.
Now we shall add 1 to N.
"N+1" will not be divisable by any of the other primes since it is not a multiple of the primes that were given. Hence N + 1 is a Prime.
But N + 1 is also larger than any of the primes that were given, thus it contradicts our first assumption of the list given was complete. Therefore the list will have no end.