Applying AM-GM to the terms $x_{1}=1, x_{2}=2, \ldots, x_{n}=n$, we have
$$
\begin{aligned}
\frac{1+2+\cdots+n}{n} & \geq \sqrt[n]{1 \times 2 \times \cdots \times n} \\
\frac{\frac{n(n+1)}{2}}{n} & \geq \sqrt[n]{n !} \\
\left(\frac{n+1}{2}\right)^{n} & \geq n !
\end{aligned}
$$
Since $n>1$, the terms $x_{i}$ are not all equal, and hence equality does not hold. As such, this shows that
$$
\left(\frac{n+1}{2}\right)^{n}>n ! . \square
$$