Suppose that for some number n there is no prime between n and n!.

Now consider (n!-1), this is not divisible by any number 2<=k<=n, and so

is not divisible by any prime <=n. Hence either (n!-1) is prime or there is

some prime q, n<n<n!, which divides (n!-1). Either case is a contradiction

and so there must be a prime netween n and n!.

RonL