Euclid's proof that there are infinitely many primes is short. Suppose the list of primes ended. Multiply them all and add one. That number is not divisible by any prime on the list, so either it is a new prime or it has a prime factor you missed. Either way the list was not finished. There is no last prime. The shelf does not close.
Worth knowing
There is no largest prime. Euclid's argument is a one-page proof of an endless list.