Proving by induction that $n! < (\frac{n+1}{2})^n$

By hypothesis, we have

$$\begin{align} (n+1)!&=(n+1)n!\\\\ &<(n+1)\left(\frac{n+1}{2}\right)^n\\\\ &=2\left(\frac{n+1}{2}\right)^{n+1}\end{align}$$

From Bernoulli's Inequality, we find that

$$\begin{align} \left(\frac{n+2}{2}\right)^{n+1} &=\left(\frac{n+1}{2}\right)^{n+1}\left(1+\frac{1}{n+1}\right)^{n+1}\\\\ &\ge 2\left(\frac{n+1}{2}\right)^{n+1} \end{align}$$

And we are done!