KöMaL 2007 B4033

All the participants in a dance course are married couples. Someone noticed that the difference between the heights of the two members of each couple is at most 10 cm. The dance instructor asks both the men and the women to stand in a line in increasing order of heights. Dancing couples are made by this rule: The tallest woman dances with the tallest man, and so on. Prove that the differences between the members of every dancing couple are also at most 10 cm.