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    Characterize Real Numbers

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    Characterize the set of all real numbers with the discrete metric as to whether it is compact, complete, or totally bounded. Use definitions only! (i.e. compact => every sequence converges, etc)

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    Let R is the set of all real numbers. We define the discrete metric as d(x,y)=0 if x=y and d(x,y)=1 if x<>y. Here "<>" denotes "not equal to". By this definition, we have
    (a) R is not compact.
    Compact means that every sequence in R has a convergent subsequence. But we consider N={1,2,3,...}, the set of ...

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    This shows how to characterize the set of real numbers.

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