assuming the fact that every infinite set contains a denumerable subset, prove the following: every infinite set is equivalent to one of its proper subset.
(hint: prove this first for a denumerable set a, i. e. prove that there is an injective function f: a? a that is not surjective. then the range would be a denumerable proper subset of a. for a general infinite set b which contains a denumerable subset a, define a function g: b? b such that g|a=f and g|ac=idac. is gg injective? is g surjective? )