Prove that there is a prime number p such that n < p < n! for all n <2. n is an integer.
I'm not allowed to use Prime number theorem or this kind of stuff. Should be simple, but you must think of it!
Let be the greatest prime, less or equal than , then can only be divisible by primes greater than ( see here), thus,let be a prime number satisfying then (1) ( since otherwise we'd contradict our definition of )
From it follows that if ( show by induction that for all )
Thus, since we have (2)
(1) and (2) complete the proof for
Suppose to the contrary there is no prime with
Let be all the primes less than or equal to and consider
is greater than 1 and not divisible by any of the primes hence must be divisible by a prime
It follows that But say
Hence This can only hold if the LHS is 0, i.e. This is impossible for (contradiction).