Originally the problem is to prove that $n! \geq n^{n/2}$.
I reduced this to: $n! \geq (\sqrt{n})^n$ so that:
Prove that $\frac{n!}{(\sqrt{n})^n} \geq 1$.
Each term in $n!$ is divided by the $\sqrt{n}$ and the multiplication should leave it $\geq 1$.
Some advice.