M = 1 + 2·3·5···PL = 1 + ∏i=1L PiIt follows that M is not evenly divisible by any prime, as M/Pi gives a remainder of 1 for all of the primes from 2 to PL, inclusive. That is, M is prime and M > PL, a contradiction, and our result is established. QED
Note: The above proof is attributed to Euclid (circa 300 BC). Another proof, due to Euler, states that the sum of the reciprocals of all prime numbers diverges.