Show that every uncountable set of real numbers has a point of accumulation.
Show uncountable set of real numbers has a point of accumulation
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real-analysis
general-topology
analysis
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11Okay. I've shown that. Oh, you wanted me to post an answer too? – 2012-10-22
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0You may be interested in [this](http://dantopology.wordpress.com/2010/05/29/the-lindelof-property-of-the-real-line/) web page. – 2012-10-22
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3Instead of just *demanding* us to show something, how about stating what your *own* efforts have been so far? – 2012-10-22
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0Related: [Accumulation points of uncountable sets](http://math.stackexchange.com/questions/310113/accumulation-points-of-uncountable-sets) – 2015-10-22
1 Answers
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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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0It 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? – 2016-02-22
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0@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. ' – 2017-08-07