Intro to Analysis in-class_Part_25

# Intro to Analysis in-class_Part_25 - 3.3 Compact sets 55...

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Unformatted text preview: 3.3 Compact sets 55 Corollary 3.3.7. Any infinite subset of a compact set has a cluster point. Proof. This comes from the two previous theorems Definition 3.3.8. An open cover of A ⊆ R is a collection of open sets { U i } with A ⊆ S U i . Example 3.3.1. (0 , 1) ⊆ (0 , 1) ⊆ (- 1 , 2). (0 , 1) ⊆ (0 , 2 3 ) ∪ ( 1 3 , 1). (0 , 1) ⊆ S n ( 1 n , 1- 1 n ). R ⊆ S n (- n,n ). U ⊆ S x ∈ U ( x- ε,x + ε ). Theorem 3.3.9. Let K ⊆ R be compact, and let B be a closed subset of K . Then B is compact. Proof. K is closed and bounded, by thm, so B ⊆ K must also be bounded. Since B is closed by hypothesis, B is compact. Theorem 3.3.10 (Heine-Borel Thm) . A set is compact iff every open cover has a finite subcover. Proof. ( ⇒ ) Suppose A is an open cover of a compact set K . First, reduce A to a countable subcover B . Let I be any open interval with rational endpoints. If there is an open set of A which contains I , then add it to B . If not, then don’t. Now any point of K is contained...
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Intro to Analysis in-class_Part_25 - 3.3 Compact sets 55...

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