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?

we have to prove that $(n+1)!\le (n+1)^{n+1}$ now we get $(n+1)!\le (n+1)\cdot n^n$ and it is easy to show that $(n+1)\cdot n^n \le (n+1)^{n+1}$ the last inequality is true since $n\le n+1$
