# Prove: Set Theory, closed sets and compact sets

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)

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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.

Proof:

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.

$2.19