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This text originated from the lecture notes I gave teaching the honours undergraduate-level real analysis sequence at the University of Califor- nia, Los Angeles, in 2003. Among the undergraduates here, real anal- ysis was viewed as being one of the most difficult courses to learn, not only because of the abstract concepts being introduced for the first time (e.g., topology, limits, measurability, etc.), but also because of the level of rigour and proof demanded of the course. Because of this percep- tion of difficulty, one was often faced with the difficult choice of either reducing the level of rigour in the course in order to make it easier, or to maintain strict standards and face the prospect of many undergradu- ates, even many of the bright and enthusiastic ones, struggling with the course material. Faced with this dilemma, I tried a somewhat unusual approach to the subject. Typically, an introductory sequence in real analysis assumes that the students are already familiar with the real numbers, with math- ematical induction, with elementary calculus, and with the basics of set theory, and then quickly launches into the heart of the subject, for in- stance the concept of a limit. Normally, students entering this sequence do indeed have a fair bit of exposure to these prerequisite topics, though in most cases the material is not covered in a thorough manner. For in- stance, very few students were able to actually define a real number, or even an integer, properly, even though they could visualize these num- bers intuitively and manipulate them algebraically. This seemed to me to be a missed opportunity. Real analysis is one of the first subjects (together with linear algebra and abstract algebra) that a student en- counters, in which one truly has to grapple with the subtleties of a truly rigorous mathematical proof. As such, the course offered an excellent chance to go back to the foundations of mathematics, and in particular xiii xiv Preface to the first edition the opportunity to do a proper and thorough construction of the real numbers. Thus the course was structured as follows. In the first week, I de- scribed some well-known “paradoxes” in analysis, in which standard laws of the subject (e.g., interchange of limits and sums, or sums and inte- grals) were applied in a non-rigorous way to give nonsensical results such as 0 = 1. This motivated the need to go back to the very beginning of the subject, even to the very definition of the natural numbers, and check all the foundations from scratch. For instance, one of the first homework assignments was to check (using only the Peano axioms) that addition was associative for natural numbers (i.e., that (a + b) + c = a + (b + c) for all natural numbers a,b,c: see Exercise 2.2.1). Thus even in the first week, the students had to write rigorous proofs using mathematical induction. After we had derived all the basic properties of the natural numbers, we then moved on to the integers (initially defined as formal differences of natural numbers); once the students had verified all the basic properties of the integers, we moved on to the rationals (initially defined as formal quotients of integers); and then from there we moved on (via formal limits of Cauchy sequences) to the reals. Around the same time, we covered the basics of set theory, for instance demonstrat- ing the uncountability of the reals. Only then (after about ten lectures) did we begin what one normally considers the heart of undergraduate real analysis - limits, continuity, differentiability, and so forth. The response to this format was quite interesting. In the first few weeks, the students found the material very easy on a conceptual level, as we were dealing only with the basic properties of the standard num- ber systems. But on an intellectual level it was very challenging, as one was analyzing these number systems from a foundational viewpoint, in order to rigorously derive the more advanced facts about these number systems from the more primitive ones. One student told me how difficult it was to explain to his friends in the non-honours real analysis sequence (a) why he was still learning how to show why all rational numbers are either positive, negative, or zero (Exercise 4.2.4), while the non- honours sequence was already distinguishing absolutely convergent and conditionally convergent series, and (b) why, despite this, he thought his homework was significantly harder than that of his friends. Another student commented to me, quite wryly, that while she could obviously see why one could always divide a natural number n into a positive integer q to give a quotient a and a remainder r less than q (Exercise 2.3.5), she still had, to her frustration, much difficulty in writing down Preface to the first edition xv a proof of this fact. (I told her that later in the course she would have to prove statements for which it would not be as obvious to see that the statements were true; she did not seem to be particularly consoled by this.) Nevertheless, these students greatly enjoyed the homework, as when they did perservere and obtain a rigorous proof of an intuitive fact, it solidified the link in their minds between the abstract manipulations of formal mathematics and their informal intuition of mathematics (and of the real world), often in a very satisfying way. By the time they were assigned the task of giving the infamous “epsilon and delta” proofs in real analysis, they had already had so much experience with formalizing intuition, and in discerning the subtleties of mathematical logic (such as the distinction between the “for all” quantifier and the “there exists” quantifier), that the transition to these proofs was fairly smooth, and we were able to cover material both thoroughly and rapidly. By the tenth week, we had caught up with the non-honours class, and the students were verifying the change of variables formula for Riemann-Stieltjes in- tegrals, and showing that piecewise continuous functions were Riemann integrable. By the conclusion of the sequence in the twentieth week, we had covered (both in lecture and in homework) the convergence theory of Taylor and Fourier series, the inverse and implicit function theorem for continuously differentiable functions of several variables, and established the dominated convergence theorem for the Lebesgue integral. In order to cover this much material, many of the key foundational results were left to the student to prove as homework; indeed, this was an essential aspect of the course, as it ensured the students truly ap- preciated the concepts as they were being introduced. This format has been retained in this text; the majority of the exercises consist of proving lemmas, propositions and theorems in the main text. Indeed, I would strongly recommend that one do as many of these exercises as possible - and this includes those exercises proving “obvious” statements - if one wishes to use this text to learn real analysis; this is not a subject whose subtleties are easily appreciated just from passive reading. Most of the chapter sections have a number of exercises, which are listed at the end of the section. To the expert mathematician, the pace of this book may seem some- what slow, especially in early chapters, as there is a heavy emphasis on rigour (except for those discussions explicitly marked “Informal”), and justifying many steps that would ordinarily be quickly passed over as being self-evident. The first few chapters develop (in painful detail) many of the “obvious” properties of the standard number systems, for xvi Preface to the first edition instance that the sum of two positive real numbers is again positive (Ex- ercise 5.4.1), or that given any two distinct real numbers, one can find rational number between them (Exercise 5.4.5). In these foundational chapters, there is also an emphasis on non-circularity - not using later, more advanced results to prove earlier, more primitive ones. In partic- ular, the usual laws of algebra are not used until they are derived (and they have to be derived separately for the natural numbers, integers, rationals, and reals). The reason for this is that it allows the students to learn the art of abstract reasoning, deducing true facts from a lim- ited set of assumptions, in the friendly and intuitive setting of number systems; the payoff for this practice comes later, when one has to utilize the same type of reasoning techniques to grapple with more advanced concepts (e.g., the Lebesgue integral). The text here evolved from my lecture notes on the subject, and thus is very much oriented towards a pedagogical perspective; much of the key material is contained inside exercises, and in many cases I have chosen to give a lengthy and tedious, but instructive, proof in- stead of a slick abstract proof. In more advanced textbooks, the student will see shorter and more conceptually coherent treatments of this ma- terial, and with more emphasis on intuition than on rigour; however, I feel it is important to know how to do analysis rigorously and “by hand” first, in order to truly appreciate the more modern, intuitive and abstract approach to analysis that one uses at the graduate level and beyond. The exposition in this book heavily emphasizes rigour and formal- ism; however this does not necessarily mean that lectures based on this book have to proceed the same way. Indeed, in my own teach- ing I have used the lecture time to present the intuition behind the concepts (drawing many informal pictures and giving examples), thus providing a complementary viewpoint to the formal presentation in the text. The exercises assigned as homework provide an essential bridge between the two, requiring the student to combine both intuition and formal understanding together in order to locate correct proofs for a problem. This I found to be the most difficult task for the students, as it requires the subject to be genuinely learnt, rather than merely memorized or vaguely absorbed. Nevertheless, the feedback I received from the students was that the homework, while very demanding for this reason, was also very rewarding, as it allowed them to connect the rather abstract manipulations of formal mathematics with their innate intuition on such basic concepts as numbers, sets, and functions. Of Preface to the first edition xvii course, the aid of a good teaching assistant is invaluable in achieving this connection. With regard to examinations for a course based on this text, I would recommend either an open-book, open-notes examination with problems similar to the exercises given in the text (but perhaps shorter, with no unusual trickery involved), or else a take-home examination that involves problems comparable to the more intricate exercises in the text. The subject matter is too vast to force the students to memorize the defini- tions and theorems, so I would not recommend a closed-book examina- tion, or an examination based on regurgitating extracts from the book. (Indeed, in my own examinations I gave a supplemental sheet listing the key definitions and theorems which were relevant to the examination problems.) Making the examinations similar to the homework assigned in the course will also help motivate the students to work through and understand their homework problems as thoroughly as possible (as op- posed to, say, using flash cards or other such devices to memorize mate- rial), which is good preparation not only for examinations but for doing mathematics in general. Some of the material in this textbook is somewhat peripheral to the main theme and may be omitted for reasons of time constraints. For instance, as set theory is not as fundamental to analysis as are the number systems, the chapters on set theory (Chapters 3, 8) can be covered more quickly and with substantially less rigour, or be given as reading assignments. The appendices on logic and the decimal system are intended as optional or supplemental reading and would probably not be covered in the main course lectures; the appendix on logic is particularly suitable for reading concurrently with the first few chapters. Also, Chapter 11.27 (on Fourier series) is not needed elsewhere in the text and can be omitted. For reasons of length, this textbook has been split into two volumes. The first volume is slightly longer, but can be covered in about thirty lectures if the peripheral material is omitted or abridged. The second volume refers at times to the first, but can also be taught to students who have had a first course in analysis from other sources. It also takes about thirty lectures to cover. I am deeply indebted to my students, who over the progression of the real analysis course corrected several errors in the lectures notes from which this text is derived, and gave other valuable feedback. I am also very grateful to the many anonymous referees who made several corrections and suggested many important improvements to the text. xviii Preface to the first edition I also thank Biswaranjan Behera, Tai-Danae Bradley, Brian, Eduardo Buscicchio, Carlos, EO, Florian, G ̈okhan Gu ̈ ̧clu ̈, Evangelos Georgiadis, Ulrich Groh, Bart Kleijngeld, Erik Koelink, Wang Kuyyang, Matthis Lehmku ̈hler, Percy Li, Ming Li, Jason M., Manoranjan Majji, Geoff Mess, Pieter Naaijkens, Vineet Nair, Cristina Pereyra, David Radnell, Tim Reijnders, Pieter Roffelsen, Luke Rogers, Marc Schoolderman, Kent Van Vels, Daan Wanrooy, Yandong Xiao, Sam Xu, Luqing Ye, and the students of Math 401/501 and Math 402/502 at the University of New Mexico for corrections to the first and second editions. Terence Tao