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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, Bookend Oct 22 '15 at 23:35

This question appears to be off-topic. The users who voted to close gave this specific reason:

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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 '15 at 15:42
up vote 11 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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It seems that it is also true that if $A$ is an uncountable set of real numbers then $A\cap A'$ is nonempty. Is it true? How could I prove it? – MingusMingusMingusMingus Feb 22 at 23:10

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