Permutations and two sequences with the same cluster set

Abstract

Suppose that a (n-*an) and b (n-*bn) are sequences in a compact metric space with distance d. Suppose further that a and b have the same set of cluster points C. Von Neumann [1, p. 11-12] proved there exists a permutation 7r of the set of positive integers Z such that d(an, b,.)->O. Halmos [2] recently gave an improved proof (and the above statement for compact metric spaces). A discussion of this result may be found in [2 ]. My purpose is to give a shorter proof than that of Halinos, which used the Schroder-Bernstein Theorem. PROOF. Let 7rl = 1. We now construct 7rn inductively for n > 1, given 7r , * * * I 7r(n-1). Write Z, for Z and Zn for Z{7rl, II .* . j(n) } . Let p(n) = min Zn. Let 7rn be the smallest integer in Zn such that