BALTIC-WAY 1993 Problem 10

Let $a_{1}, a_{2}, \ldots, a_{n}$ and $b_{1}, b_{2}, \ldots, b_{n}$ be two finite sequences consisting of $2 n$ different real numbers. Rearranging each of the sequences in the increasing order we obtain $a_{1}^{\prime}, a_{2}^{\prime}, \ldots, a_{n}^{\prime}$ and $b_{1}^{\prime}, b_{2}^{\prime}, \ldots, b_{n}^{\prime}$. Prove that

$$\max _{1 \leq i \leq n}\left|a_{i}-b_{i}\right| \geq \max _{1 \leq i \leq n}\left|a_{i}^{\prime}-b_{i}^{\prime}\right| .$$