Let $ x_1,x_2,\dots,x_n$ be positive real numbers. Let $ m=\min\{x_1,x_2,\dots,x_n\}$, $ M=\max\{x_1,x_2,\dots,x_n\}$, $ A=\frac{1}{n}(x_1+x_2+\dots+x_n)$, and $ G=\sqrt[n]{x_1x_2 \dots x_n}$. Prove that \[ A-G \ge \frac{1}{n}(\sqrt{M}-\sqrt{m})^2.\]