Prove that every uncountable subset of $R$ (real numbers), has a limit point.
I tried using Baire Category Theorem, which deals with uncountability, but I'm at sea.
If anyone can please help me with this problem I'll be glad. Thanks in advance
Question on Real Numbers and Limit Point
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real-analysis
metric-spaces
2 Answers
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Write $\mathbb{R} = \bigcup\limits_{n=-\infty}^\infty [n,n+1).$ Then think about whether every term in this union could have only finitely many points of the uncountable set in question, and what happens if one of them has more than finitely many.
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0this proof is so fantastic! Thank you so much – 2011-11-16
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Suppose that $C$ is a subset of $\mathbb{R}$ with no limits point. Then for every $x\in\mathbb{R}$ you can find rational numbers $p_x$ and $q_x$ such that $p_x