 Juan Pablo Pinasco has written the following proof.Let "p"1, ..., "p""N" be the smallest "N" primes.Then by the inclusion–exclusion principle, the number of positive integers less than or equal to "x" that are divisible by one of those primes is

Dividing by "x" and letting "x" → ∞ gives

This can be written as

If no other primes than "p"1, ..., "p""N" exist, then the expression in (1) is equal to formula_13 and the expression in (2) is equal to 1, but clearly the expression in (3) is not equal to 1.