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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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 Fall '11
 Wong
 Sets

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