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

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    $\begingroup$ Okay. I've shown that. Oh, you wanted me to post an answer too? $\endgroup$ – Asaf Karagila Oct 22 '12 at 20:49
  • $\begingroup$ You may be interested in this web page. $\endgroup$ – David Mitra Oct 22 '12 at 21:30
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    $\begingroup$ Instead of just demanding us to show something, how about stating what your own efforts have been so far? $\endgroup$ – Hagen von Eitzen Oct 22 '12 at 21:41
  • $\begingroup$ Related: Accumulation points of uncountable sets $\endgroup$ – 6005 Oct 22 '15 at 15:42
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Hint:

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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  • $\begingroup$ 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? $\endgroup$ – JKEG Feb 22 '16 at 23:10
  • $\begingroup$ @Asaf Karagila sir . Can you prove how this is coming If $A$ is an uncountable set of real numbers then there exists $ k∈Z$ such that $A∩[k,k+1$] is infinite. ' $\endgroup$ – Unknown x Aug 7 '17 at 2:50

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