I got some question on how to proceed on the proof below,
Prove that:
$n^{n+1}>(n+1)^{n}$, for $n\geq3$
By induction:
Inequality holds for $n=3$ , $3^4=81\geq 4^3 =64$.
Suppose it holds for $k^{k+1}>(k+1)^{k}$.
Prove for $k+1$ :
$(k+1)^{k+2}\geq(k+2)^{k+1}$
and here is the part where I am kind of stuck, how would I use the hypothesis to prove for $k+1$?
Your inequality is equivalent to
$$n> \left(1+\frac1n\right)^n.$$
Assuming it holds for some $n\ge 3$ we have to show it for $n+1.$ We have:
$$\left(1+\frac1{n+1}\right)^{n+1}< \left(1+\frac1n\right)^{n+1}=\left(1+\frac1n\right)\left(1+\frac1n\right)^n< \left(1+\frac1n\right)n=n+1,$$ what finishes the proof.