BabyRudin: Solution: Ex15 in Ch02

Books/BabyRudin/Chapter02/ex15.tex

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+\subsection*{Exercise 15}
+To show that the theorem is not true for closed subsets, consider the family of set $ \overline{\{n\}} $ for $ n \in \mathbb{N} $ as subsets of the real number line.
+
+All the sets are closed because these isolated points have no additional limit points. (i.e any points that is not a natural number will have a neighborhood that does not intersect with the set.)
+Any finite intersection is not empty there is a maximum number that is being deleted by the sets, and the number after it is not deleted by any set and therefore belong to the intersection.
+But the infinite intersection is empty because there is no maximum number that is not deleted by any set.
+
+To show that the theorem is not true for bounded subsets, consider the family of set $ [1, 2] $ with $ r $ deleted for all real numbers $ r $.
+
+All the sets are bounded because they are all subsets of $ [1, 2] $.
+Any finite intersection is not empty because there are infinitely many real numbers in $ [1, 2] $ but we only deleted a finite number of them, so there must be a number that is not deleted.
+But the infinite intersection is empty because we deleted all the real numbers in $ [1, 2] $.
+