Induction proof of exponential and factorial inequality I'm trying to find a proof for the following statement, using mathematical induction:
$$ (\forall n\in \mathbb N-\{0\}) n^n \ge n! $$
But I always get to a dead-end.
I've done the basis step, for $n = 1$, which is clearly true, but I can't prove for an arbitrary $k+1$ if I assume it's true for $k$.
How can I prove this?

You can use
$$(k+1)!=(k+1)\cdot k!$$
and
$$k^k\le (k+1)^k.$$
as the following :

 $(k+1)!=(k+1)\cdot k!\le (k+1)\cdot k^k\le (k+1)\cdot (k+1)^k=(k+1)^{k+1}.$

