How to solve the inequality $n! \le n^{n-2}$? The inequality is $n! \le n^{n-2}$. I used Stirling's approximation for factorials and my answer was $n \le (e(2\pi)^{-1/2})^{2/5}$ but this doesn't seem right. Any help would be much appreciated.

Here's a proof by induction. 
First, we need to verify the base case: $5!=120 < 125=5^3$
Now assume the induction hypothesis $n! < n^{n-2}$ is true, then we need to prove 
$$(n+1)! < (n+1)^{n-1} \Leftrightarrow n! < (n+1)^{n-2}.$$
But by induction hypothesis and the fact $n < n+1$ we have,
$$n! < n^{n-2} < (n+1)^{n-2}.$$
Therefore $(n+1)! < (n+1)^{n-1}$ and we are done. 
