How can I show that $n! \leqslant \left(\frac{n+1}{2}\right)^n$?

Use the AM-GM inequality on the numbers $1,...,n$.

Hint: $$ (n!)^2 = 1\times n \times 2\times (n-1) \times \dots = \prod_{k=1}^n k(n+1-k) $$ then use $$ k\times (n+1 -k) = \left(\frac {n+1}2 + \frac {n+1}2 - k\right)\left(\frac {n+1}2 - \frac {n+1}2 + k\right) \\= \left(\frac {n+1}2\right)^2 - \left( \frac {n+1}2 - k\right)^2 \le \left(\frac {n+1}2\right)^2 $$ to get $$ (n!)^2 \le \left(\frac {n+1}2\right)^{2n} $$

HINT:

$$\frac{r+n+1-r}2\ge\sqrt{r(n-r)}$$ for $1\le r\le n$

Set $r=1,2,\cdots,n-1,n$ and then multiply