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    Prove Compactness of a Closed Subset

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    Prove that any closed subset of compact metric space is compact by using Theorem 2.

    Theorem 2: A subset of S of a metric space X is compact if, and only if, every sequence in S has a subsequence that converges to a point in S.

    © BrainMass Inc. brainmass.com October 10, 2019, 7:04 am ad1c9bdddf
    https://brainmass.com/math/calculus-and-analysis/prove-compactness-closed-subset-566847

    Solution Preview

    Proof:
    We consider a closed subset S in a compact metric space C. Then we consider any sequence {x_n} in S. Since {x_n } ...

    Solution Summary

    The solution proves a closed subset of a compact metric space is also compact.

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