1st the problem: n! is never a perfect square for all n element of natural numbers.

The way I tried to attack it seems to work but it doesnt develop a proof. I noted that n! = n(n-1)...(2)(1) and sqrtn! is sqrt of all those individual terms and there will always be an irrational number in that set. Therefore it could never be a perfect square because an irrational number times anything but itself or its inverse gives an irrational number. My problem is i dont know how to prove there will always be an irrational number in that set. I looked at it broken out for up to like 15! and the number of irrational factors seems to be getting larger but that doesnt constitute proof.

Am I just wasting my time is there an easier way to go about this?