Permutations of sequences and the Schröder-Bernstein theorem

Abstract

Suppose that a (nan) and b (n->bn) are sequences in a compact metric space with distance d. If d(an, bn)->O, then, clearly, a and b have the same set of cluster points. If, more generally, 7r is a permutation of the set of positive integers, such that d(an, b,,n)->O, then, again, a and b have the same set of cluster points. It is a remarkable result due to J. von Neumann (Charakterisierung des Spektrums eines Integraloperators, Hermann, Paris, 1935, pp. 11-12) that the converse is true: if two sequences a and b have the same set of cluster points, then there exists a permutation 7r of the set of positive integers suchthat d(a., b7,)>O. (In fact von Neumann discussed real sequences only, but that is merely a notational specialization.) If the common cluster set is a singleton, the permutation can be chosen to be the identity and the conclusion is immediate. If, more generally, the common cluster set is finite, the result is less trivial but still quite easy. The proof von Neumann gave for the general case is a page densely packed with subscripts. The purpose of this note is to give a simpler proof. The simplification is achieved by use of the best known result in infinite combinatorics, the Schroder-Bernstein theorem. An a posteriori analysis of von Neumann's proof shows that the subscripts hide a re-proof of that theorem in the special case at hand. If C is the common cluster set of a and b, writee, = 1e/n+d(a., C), and let U7, be the open ball with center an, and radius en,. Since e. > d(a., C), the ball U7, contains at least one point of C, and, consequently, U7, contains infinitely many terms of the sequence b. Let o1 be the smallest positive integer such that 1 > 1 and b,l C U1. Inductively, let o(n+1) be the smallest positive integer such that o(n+1) >o(n) and b,(n+l) E U+1. (Observe that o(n) >n.) Since en -30 it follows that d(a7, bn,)>O. Summary: there exists a one-toone mapping oof the set of positive integers into itself such that o-n>n for all n and such that d(an, ban)->O. Similarly, there exists a one-to-one mapping i of the set of positive integers into itself such that -n > n for all n and such that d (a7n, bn)->O. The Schroder-Bemstein theorem says that if M and N are sets and if c: M->N and r: NM are injections, then there exists a bijection