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Prove or disprove the following.
a) If S is a set of real numbers and is a compact set of real numbers, then is compact.
b) Every finite set of real numbers is compact.
c) Let be a set of real numbers and let be an open cover of consisting of an infinite number of open sets, no one of which contains . If contains a finite subcover of , then is compact.
d) If the intersection of an infinite collection of open sets is nonempty, then this intersection is not a compact set.
Let and be disjoint nonempty compact sets.
is compact. Let be a sequence such that for every . Because of Bolzano-Weierstrass Theorem 3.4.8. we know that there exists a subsequence and since is closed it follows that the limit of is in .
In other words, every sequence in has a subsequence that converges to a point that is in . Now, we know that is bounded wich means that it has an infimum and supremum.
Let and and lets observe their difference:
As and are bounded it follows that is also bounded a therefore has infimum. We have:
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