We present here a short proof of Cantor-Berstein-Schroeder theorem based on a fixed-point argument, probably not known enough; this proof is less explicit that the usual one, but it is worth noticing that it does not depend on the axiom of choice.
As a lemma, we first prove a particular case of Knaster-Tarski fixed point theorem:
Theorem: (Tarski) Let be a set and be a nondecreasing function (ie. such that for all ). Then has a fixed point.
Proof. Let and .
Notice that for all , , hence .
Moreover, implies ie. , so .
We deduce that is a fixed point of from and , and the theorem follows.
Theorem: (Cantor-Bernstein-Schroeder) Let be two sets. Suppose that there exist two injections and . Then there exists a bijection .
Proof. First, we define the map
Easily, we see that is nondecreasing, so has a fixed point according to Tarski’s theorem. Now notice that
and that induces a bijection and induces a bijection . Therefore, and are equipotent. By the way, an explicit bijection from to is given by