For i and j positive integers, let Rij be the rectangle with vertices at (0, 0), (i, 0), (0, j), (i, j). Show that any infinite set of Rij must have two rectangles one of which covers the other.
Solution
Take any rectangle Rij. There are only finitely many rectangles Rmn with m ≤ i and n ≤ j. So there must be infinitely many with m > i or infinitely many with n > j. wlog we may assume there are infinitely many with m > i. They must all have n < j or they would contain Rij. But there are only finitely many values for n available < j, so infinitely many must share some value n' < j. But now given any two of these, one contains the other.
© John Scholes
jscholes@kalva.demon.co.uk
1 Nov 2003
Last corrected/updated 1 Nov 03