RealAnalysisNotes_0114.pdf - Notes on Real Analysis 3A03...

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Notes on Real Analysis, 3A03Prof. S. Alama1Revised January 15, 2020IntroductionReal Analysis is the study of the real numbersRand of real valued functions of one or morereal variables.If this sounds familiar, it should: you’ve been doing calculations with realnumbers and functions for several years now, starting in high school and continuing throughcalculus.On the other hand, most of what you’ve done with real numbers and functionshas involved applying “rules” intended to justify which kinds of calculations are appropriate(in the sense that they lead to correct results,) which might have seemed mysterious andarbitrary to you at the time.The goal of this course is to develop some understandingof functions of a real variable using logical deductive reasoning, and in doing so obtain amuch firmer grasp of how these “rules” may be justified and how to determine whether anycomputational procedure will always yield correct results.One of the great mysteries, in fact, is to figure out what the real numbers actuallyare.You’ve learned that they include all of the other more familiar number systems from elemen-tary mathematics: the counting (natural) numbersN, integersZ, rational numbers (fractionsof integers)Q, and algebraic numbers (ie, roots of polynomials with integer coefficients, suchasp2,3p5, . . ..) In high school you mostly learned thealgebraof real numbers. Amazingly,one cannot really understand the reals via algebra.Their fundamental properties involveanalysis– basically, one must understand some sense oflimits.Let’s summarize some algebraic and geometric properties of the reals, which you aleadyknow but perhaps haven’t thought of in these terms:Ris afield:Ifa, b, c2R, then so area+b,ab, andb/cifc6= 0.Addition andmultiplication are commutative,a+b=b+aandab=ba, associative,a+ (b+c) =(a+b) +canda(bc) = (ab)c, and distributive,a(b+c) =ab+ac. Eacha2Rhas anadditive inverse-a2R, and a multiplicative inverse 1/a2R, provideda6= 0.Ris anorderedset. Ifa6=bare distinct real numbers, then eithera < borb < a.In addition, the ordering is transitive (ie, ifa > bandb > c, thena > c,) and isconsistent with the algebraic operations. In particular, ifa, b, c2Rwitha < bthena+c < b+c. Ifc >0, thenac < bc, but ifc <0, thenac > bc.1c 2018 All Rights Reserved. Do not distribute without author’s permission.1
Rhas a “metric property”, determined by the absolute value,|x|=(x,ifx0,-x,ifx <0.Then, thedistancebetweena, b2Ris defined byd(a, b) =|a-b|, the length of theline segment joining the two points on the number line.We sayRis anordered field.However, the rational number setQis also an orderedfield! The distinction betweenRandQ, and the intimate relationship between them forman important theme in the course.To see how the algebraic and ordering properties aredefined, and how the various familiar “rules” are justified from the axioms, see Section 2 inthe Bartle & Sherbert textbook.

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