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**particlejohn** Prove that if a set $\displaystyle S $ is countably or uncountably infinite, then there is a proper subset $\displaystyle T \subset S $ and a bijection $\displaystyle f: S \to T $.

Consider the case in which the set is countably infinite. Then $\displaystyle \text{card}(S) = \aleph_0$. Then there is a set $\displaystyle T $ such that $\displaystyle \text{card}(T) < \text{card}(S) $. And so there is an injective function $\displaystyle f: T \to S $. Then $\displaystyle f: S \to T $ is an injection and also a surjection which implies that it is a bijection.

Consider case in which $\displaystyle S $ is uncountably infinite. Then $\displaystyle \text{card}(S) > \aleph_{0} $. Suppose that $\displaystyle T $ is a set such that $\displaystyle \text{card}(S) > \text{card}(T) > \aleph_{0} $.

Now how do I proceed?