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# INTRODUCTION TO ANALYSIS I MATH 321

Rice University

GPA 3.86

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This 2 page Class Notes was uploaded by Jayde Lang on Monday October 19, 2015. The Class Notes belongs to MATH 321 at Rice University taught by Stephen Semmes in Fall. Since its upload, it has received 14 views. For similar materials see /class/224928/math-321-rice-university in Mathematics (M) at Rice University.

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Date Created: 10/19/15

Math 321 Handout 8 If 17 b are real numbers with a g b then the closed interval 07b from a to b is the set of real numbers 90 such that a g 90 g b The length of this interval is b 7 a Closed intervals are clearly closed subsets of the real line with respect to the standard metric Suppose that 17127 is a sequence of closed intervals in the real line such that 71 Q I for every j Z 1 Let aj b be the endpoints of 7 so that aj g b in particular The hypothesis that ajdrhbj l Q aj7bj implies that aj g aj1 and bi g b for every j Z 1 This implies in turn that aijl for all positive integers j l For ifj g l7 then aj g a g bl and ifj 2 l7 then aj g b g bl Let A be the set of the as Since aj g b1 for every j Z 17 A has an upper bound7 and hence a suprernurn Each E is an upper bound for A7 and thus supA g E for every l 2 1 Of course the suprernurn of A is an upper bound for A7 which is to say that aj g supA for every j Z 1 It follows that supA E I for every j Z 1 Math 321 Handout 8b Fix a positive integer 71 If 11 an b1 bn are real numbers such that a g b for i 17 n then the corresponding cell is the set C Q R consisting of ac x1 xn such that a g 90 g b 1 g i g n One can show that cells satisfy the limit point property and are compact as subsets of R equipped with the Euclidean metric in much the same way as for closed intervals in the real line There are two main changes in the argument as follows First if C is a cell in R associated to the real numbers 11 an b1 bn as above then the diameter of C or maximal distance between two elements of C is equal to 7L 7 102 i1 Just as an interval is the union of two subintervals of half the length a cell C Q R is the union of 2 cells of one half the diameter of C Second if C1C2 is a sequence of cells in R such that CH1 Q C for every j Z 1 then there is a t E R such that t E Lil Cj This is basically the same as the one dimensional case repeated 71 times for the 71 components of t

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