I can just figure out that arbitrary uncountable set has a countable subset, which is trivial. However, I don't know whether I can get a bounded one.
Does every uncountable subset of $\mathbb R$ have a bounded countable subset?
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analysis
elementary-set-theory
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1If we reject the Axiom of Choice, then it is possible that there is an infinite set of real numbers that has no countable subset. – 2012-09-21
1 Answers
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Yes, you can. You can also get a bounded uncountable subset.
Let $X$ be your uncountable subset of $\Bbb R$. For each integer $n$ let $X_n=X\cap[n,n+1)$. Then $X=\bigcup_{n\in\Bbb Z}X_n$, and a countable union of countable sets is countable, so at least one of the sets $X_n$ must be uncountable. That $X_n$ is then a bounded, uncountable subset of $X$. Now just pick a countably infinite subset of $X_n$, and you have your bounded, countably infinite subset of $X$.