Prove that if a set is countably or uncountably infinite, then there is a proper subset and a bijection .

Consider the case in which the set is countably infinite. Then . Then there is a set such that . And so there is an injective function . Then is an injection and also a surjection which implies that it is a bijection.

Consider case in which is uncountably infinite. Then . Suppose that is a set such that .

Now how do I proceed?