Published Jun 27, 2021
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Definition 3.3.1 (Compactness). A set \(K \subseteq \mathbb{R}\) is compact if every sequence in \(K\) has a subsequence that converges to a limit that is also in \(K\).
Definition 3.3.3 A set \(A \subseteq \mathbb{R}\) is bounded if there exists \(M > 0\) such that \(|a| \leqslant M\) for all \(a \in A\)
Theorem 3.3.4 (Characterization of Compactness in R). A set \(K \subseteq \mathbb{R}\) is compact if and only if it is closed and bounded.
Theorem 3.3.5 (Nested Compact Set Property). If $$ K_1 \supseteq K_2 \supseteq K_3 \supseteq K_4 \supseteq \cdot \cdot \cdot $$ is a nested sequence of nonempty compact sets, then the intersection \(\bigcap_{n=1}^\infty K_n\) is not empty
Definition 3.3.6 Let \(A \subseteq \mathbb{R}\). An open cover for \(A\) is a (possibly finite) collection of open sets \(\{O_\lambda : \lambda \in \Lambda\}\) whose union contains the set \(A\); that is, \(A \subseteq \bigcup_{\lambda \in \Lambda} O_\lambda\). Given an open cover for \(A\), a finite subcover is a finite subcollection of open sets from the original open cover whose union still manages to completely contain \(A\).
Theorem 3.3.8 (Heine-Borel Theorem).
Let \(K\) be a subset of \(\mathbb{R}\). All of the following statements are
equivalent in the sense that any one of them implies the two others: