Prove that $(n+1)^{n-1}How would one prove that  $$(n+1)^{n-1}<n^n \ \forall n>1$$
I have tried several methods such as induction.

My try to prove $\forall n>1 : (n+1)^{n-1} < n^n$:
$$(n+1)^{n-1} < n^n \Leftrightarrow (n+1)^{n} < n^n (n+1) \Leftrightarrow \left(\frac{n+1}{n}\right)^n < n+1$$
Now apply induction on this result:
Base case: $n = 2 \Rightarrow \left(\frac{3}{2}\right)^2 < 3 \Rightarrow 2.25 < 3$ OK
Induction hypothesis: assume $\left(\frac{n+1}{n}\right)^n < n+1$ true for all $n \leq k$
Induction step: we prove that $\left(\frac{n+1}{n}\right)^n < n+1$ is true for $n = k+1$
$$\left(\frac{k+2}{k+1}\right)^{k+1} < k+2 \Leftrightarrow \left(\frac{k+2}{k+1}\right)^k < k+1$$
Now we have that $\left(\frac{k+1}{k}\right)^k > \left(\frac{k+2}{k+1}\right)^k$ and because of the induction hypothesis, we have proven by induction that $\forall n>1 : \left(\frac{n+1}{n}\right)^n < n+1$ and thus that $\forall n>1 : (n+1)^{n-1} < n^n$
