Proving $\dfrac{(n-1)^{2n-2}}{(n-2)^{n-2}}<n^n$ for $n\ge 3$

[anemone]

Prove that $\dfrac{(n-1)^{2n-2}}{(n-2)^{n-2}}<n^n$ for integer $n\ge 3$.

 

[anemone]

Spoiler: Solution of other

Recall Bernoulli's inequality, $(1+x)^t>1+tx$ when $x>-1,\,x\ne 0$ and $t\ge 1$. For $m>1$, we have

$\begin{align*}\left(\dfrac{m+1}{m}\right)^{m+1}\left(\dfrac{m-1}{m}\right)^{m-1}&=\left(1+\dfrac{1}{m}\right)^2\left(1-\dfrac{1}{m^2}\right)^{m-1}\\& >\left(1+\dfrac{1}{m}\right)^2\left(1-\dfrac{m-1}{m^2}\right)\\&=\left(\dfrac{m^3+1}{m^3}\right)\left(\dfrac{m+1}{m}\right)\\& >1\end{align*}$

Hence, $(m+1)^{m+1}>m^{2m}(m-1)^{-(m-1)}$. Setting $m=n-1$ yields the desired result.