SE-SKOLORNAS 1986 Problem 5

In the arrangement of $pn$ real numbers below, the difference between the greatest and smallest numbers in each row is at most $d$ , $d > 0$ . $$\begin{array}{l} a_{11} \, a_{12} \, ... \, a_{1n} a_{21} \, a_{22} \, ... \, a_{2n} \, . \, \, \, \, . \, \, \, \, \, \, \, \, . \, . \, \, \, \, . \, \, \, \, \, \, \, \, . \, . \, \, \, \, . \, \, \, \, \, \, \, \, . a_{n1} \, a_{n2} \, ... \, a_{nn} \end{array}$$ Prove that, when the numbers in each column are rearranged in decreasing order, the difference between the greatest and smallest numbers in each row will still be at most d.