Is it true that for any uncountable subset T of $\mathbb R$, one can find a subset S of T such that S is countable. If yes, how can we prove it?
Thanks!
Edit: Is there a countable subset S of T such that for every element $t\in T$, there exists $s\in S$ such that $s\geq t$?
Best Answer
Yes, every infinite set has a countably infinite subset.
However this countable subset need not be constructive (we don't have to have a nice formula defining it), because its existence requires a fragment of the axiom of choice (which allows us to make infinitely many arbitrary choices at once). Indeed it is consistent in the absence of the axiom of choice that there is an infinite set of real numbers which has no countably infinite subset (this implies that this set is uncountable).
In common, and especially introductory real analysis, the axiom of choice is assumed through and through, so you may disregard the above ramble if you wish.