Intro to Analysis in-class_Part_24

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online