Here is [roughly] Euclid's proof of this......
Let the greatest known prime = P
Now, multiply all the known primes together [including P ] and let this number = M
Now....add 1 to M.......so.....the new number is M + 1
If we divide M + 1 by any of the known primes, we get a remainder .......so...........either M + 1 itself is prime, and therefore greater than the greatest prime, P, or there exists a prime greater than any of the knowm primes which divides M and which is also greater than P
Therefore.......the list of primes is infinite