Intro to Analysis in-class_Part_24

# Intro to Analysis in-class_Part_24 - 3.3 Compact sets 53...

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Unformatted text preview: 3.3 Compact sets 53 3.3 Compact sets Definition 3.3.1. A set K ⊆ R is compact iff every sequence { x n } ⊆ K has a cluster point x ∈ K . This is an abstract version of “small” just like open was an abstract version of “big”. We will characterize compactness in R : Theorem 3.3.2. A set K ⊆ R is compact iff it is closed and bounded. but first, need two theorems. Definition 3.3.3. Let A 1 ,A 2 ,... be a sequence of sets in R . This sequence is nested iff A 1 ⊇ A 2 ⊇ ... . If this is a sequence of intervals A n = [ a n ,b n ] = { x . . . a n ≤ x ≤ b n } , then this means a n ≤ a n +1 ≤ b n +1 ≤ b n , ∀ n. Note: T ∞ n =1 A n = { x . . . x ∈ A n , ∀ n } . Theorem 3.3.4 (Nested Intervals Thm) . Suppose that A n = [ a n ,b n ] is a nested sequence of intervals with lim( a n- b n ) = 0 . Then T ∞ n =1 A n = { L } . Also, a n → L and b n → L . Proof. There are 5 steps....
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Intro to Analysis in-class_Part_24 - 3.3 Compact sets 53...

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