Show that every uncountable set of real numbers has a point of accumulation.
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14Okay. I've shown that. Oh, you wanted me to post an answer too? – Asaf Karagila Oct 22 '12 at 20:49
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You may be interested in this web page. – David Mitra Oct 22 '12 at 21:30
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4Instead 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
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Related: Accumulation points of uncountable sets – Caleb Stanford 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.
Asaf Karagila
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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? – JKEG Feb 22 '16 at 23:10
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@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. ' – Aug 07 '17 at 02:50
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2If $A \cap [k,k+1]$ was finite for all $k$, then $A =\cup_{k \in \mathbb Z} A \cap [k,k+1]$ would be countable. – Gonzalo A. Benavides Jul 14 '22 at 01:10