Induction Proof $n! < n^n$ [closed]

How would you go about proving $n! < n^n$ using a mathematical induction proof?

I understand how to solve inductive proofs with = but I'm getting a bit lost in this example.

Any help is much appreciated.

Solution 1:

First of all, we have $$ 2! < 2^2 $$ which covers the base case. Now, for the induction itself, assume we know that $k!<k^k$ for some $k$. We want to compare $(k+1)!$ and $(k+1)^{k+1}$. Note that $(k+1)! = k!\cdot (k+1)$, which lets us use the induction hypothesis (I.H.): $$ (k+1)! = k!\cdot (k+1) \overset{\text{I.H.}}{<} k^k\cdot(k+1) \overset{k<k+1}{<} (k+1)^k\cdot (k+1) = (k+1)^{k+1} $$ which is what we want.