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    Prove: Set Theory, closed sets and compact sets

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    I would like to know how to construct a proof of union/and of 2 closed sets and how to prove compact sets.

    (See attached file for full problem description)

    a. Let E and F be closed sets in R. Prove that E R is closed. Prove the E F is closed.

    b. Let E and F be compact sets in R. Prove that E F is compact. Prove that EF is not necessarily compact

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    Solution Preview

    Please see the attachment.

    We know a set is closed if and only if , where is the set of all cluster points of .
    (a) and are closed sets in .
    First, I show that is closed in . For any , we can find a sequence , such that . Since each , then or . We can select the two subsequence ...

    Solution Summary

    This solution is comprised of a detailed explanation to prove that E R is closed.