29th Putnam 1968

Problem B6

A compact set of real numbers is closed and bounded. Show that we cannot find compact sets A1, A2, A3, ... such that (1) all elements of An are rational and (2) given any compact set K whose members are all rationals, K ⊆ some An.



Use a diagonalisation argument.

Suppose we can find such An. Let Sn be the interval [1/22n, 1/22n-1], and let Tn be the set of rational points in Sn. Then the closure of Tn is Sn, which contains irrational points, so there must be points of Tn which are not in An. Let xn be one such. Now consider {xn}. Its only limit point is 0, so K, the union of {xn} and {0}, is a compact set of rationals. But K is not contained in any of the An, because it has a member xn not in An for each n. Contradiction.



29th Putnam 1968

© John Scholes
14 Jan 2002