Hi!

Could you help me prove this inequality?

$\displaystyle e^{n-1} \cdot n! < n^{n+1}$

When I try induction , I get:

$\displaystyle e^n \cdot (n+1)! = e^{n-1} \cdot n! \cdot e \cdot (n+1)<$

and this is when I have no idea how to get $\displaystyle (n+1)^{n+2}$. Could I ask for a small hint? It cannot be that hard to prove

Or maybe you have another, more constructive way to prove this inequality?