Adapt Euclid's proof.

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 .

RonL