RealAnalysisNotes_0120.pdf - Notes on Real Analysis 3A03...

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Notes on Real Analysis, 3A03 Prof. S. Alama 1 Revised January 20, 2020 Introduction Real Analysis is the study of the real numbers R and of real valued functions of one or more real variables. If this sounds familiar, it should: you’ve been doing calculations with real numbers and functions for several years now, starting in high school and continuing through calculus. On the other hand, most of what you’ve done with real numbers and functions has 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 and arbitrary to you at the time. The goal of this course is to develop some understanding of functions of a real variable using logical deductive reasoning, and in doing so obtain a much firmer grasp of how these “rules” may be justified and how to determine whether any computational procedure will always yield correct results. One of the great mysteries, in fact, is to figure out what the real numbers actually are. You’ve learned that they include all of the other more familiar number systems from elemen- tary mathematics: the counting (natural) numbers N , integers Z , rational numbers (fractions of integers) Q , and algebraic numbers (ie, roots of polynomials with integer coe ffi cients, such as p 2 , 3 p 5 , . . . .) In high school you mostly learned the algebra of real numbers. Amazingly, one cannot really understand the reals via algebra. Their fundamental properties involve analysis– basically, one must understand some sense of limits . Let’s summarize some algebraic and geometric properties of the reals, which you aleady know but perhaps haven’t thought of in these terms: R is a field : If a, b, c 2 R , then so are a + b , ab , and b/c if c 6 = 0. Addition and multiplication are commutative, a + b = b + a and ab = ba , associative, a + ( b + c ) = ( a + b ) + c and a ( bc ) = ( ab ) c , and distributive, a ( b + c ) = ab + ac . Each a 2 R has an additive inverse - a 2 R , and a multiplicative inverse 1 /a 2 R , provided a 6 = 0. R is an ordered set. If a 6 = b are distinct real numbers, then either a < b or b < a . In addition, the ordering is transitive (ie, if a > b and b > c , then a > c ,) and is consistent with the algebraic operations. In particular, if a, b, c 2 R with a < b then a + c < b + c . If c > 0, then ac < bc , but if c < 0, then ac > bc . 1 c 2018 All Rights Reserved. Do not distribute without author’s permission. 1
R has a “metric property”, determined by the absolute value, | x | = ( x, if x 0, - x, if x < 0. Then, the distance between a, b 2 R is defined by d ( a, b ) = | a - b | , the length of the line segment joining the two points on the number line. We say R is an ordered field. However, the rational number set Q is also an ordered field! The distinction between R and Q , and the intimate relationship between them form an important theme in the course. To see how the algebraic and ordering properties are defined, and how the various familiar “rules” are justified from the axioms, see Section 2 in the Bartle & Sherbert textbook.

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