For each prime let be the greatest such that .

Then

But remember that thus: .

Now: - equality holds on the RHS iff n = 1 -

The right inequality is simple enough, each factor is less than 1 - and there are no factors only if n = 1-, now for the left inequality for each , and when -since n can't have all the primes as divisors, it follows that some inequality must be strict.