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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

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2 Answers 2

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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this proof is so fantastic! Thank you so much – AdityaGhosh Nov 16 '11 at 18:03

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 $p_x<x<q_x$ and $(p_x,q_x)\cap C$ contains at most one point. (Why?) How many intervals are there with rational endpoints? So how big can $C$ be?

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