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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0Related: [Accumulation points of uncountable sets](http://math.stacke$x$ch$a$nge.com/questions/310113/accumulation-points-of-uncounta$b$le-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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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