MathNet 2019-part-01 — 184 problems (EN–VI) Question 173

Two collections $(x_1, x_2, \dots, x_n)$ and $(y_1, y_2, \dots, y_n)$ of real numbers satisfy the inequality $x_i \ge y_j$ for all indices $i$ and $j$. Let $P = \max_{1 \le i \le n}(x_i - y_i)$ and $G = \max_{1 \le i \le n} x_i - \min_{1 \le j \le n} y_j$.
Prove the inequality $P \le G \le nP$.