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Show that every uncountable set of real numbers has a point of accumulation.

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closed as off-topic by 6005, zhoraster, Hagen von Eitzen, Servaes, Normal Human Oct 22 at 23:35

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  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – 6005, zhoraster, Hagen von Eitzen, Servaes, Normal Human
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Okay. I've shown that. Oh, you wanted me to post an answer too? – Asaf Karagila Oct 22 '12 at 20:49
You may be interested in this web page. – David Mitra Oct 22 '12 at 21:30
Instead of just demanding us to show something, how about stating what your own efforts have been so far? – Hagen von Eitzen Oct 22 '12 at 21:41
Related: Accumulation points of uncountable sets – 6005 Oct 22 at 15:42

1 Answer 1

up vote 10 down vote accepted


If $A$ is an uncountable set of real numbers then there exists $k\in\mathbb Z$ such that $A\cap[k,k+1]$ is infinite. Use the definition of compactness, and the fact $[k,k+1]$ is a closed and bounded interval.

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