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
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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.