Suppose that for some there is no prime such that .
Now consider this is either a prime or composite the first of
these possibilities contradicts our assumption, so it must be composite.
Now let be the primes less than , then clearly .
But is composite, so it has a prime divisor, which therefore must be greater
than and less than , a contradiction.
So we conclude that it is not the case that that there are no primes such
that , that is for all there is a prime such that .