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    Cauchy Sequence and Completeness of a Metric Space

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    Let X be a complete metric space. If F_n is a sequence of nonempty closed subsets of X such that F_n+1 is contained in F_n and the limit as n-->infinity of the diameter(F_n) = 0, show that the interesection of all F_n is nonempty.

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    Solution Preview

    First, let's write down the relevant definitiions.
    A metric space X is complete if every Cauchy sequence {x_n} of elements of X has a limit x also in X.
    A sequence x_n is Cauchy, if for every epsilon>0 there's a number N such that for all n,m>N we have
    d(x_n , x_m) < epsilon
    here d(x,y) is the metric on X.
    Now, the diameter diam(F) of a set F in a metric space is sup {d(x,y), x and y are in F}.

    What we want to do is to construct a Cauchy ...

    Solution Summary

    Cauchy sequence is clearly evaluated in this case.