Undergraduate Texts in Mathematics Charles C. Pugh Real Mathematical Analysis Second Edition Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: ColinAdams,WilliamsCollege DavidA.Cox,AmherstCollege PamelaGorkin,BucknellUniversity RogerE.Howe,YaleUniversity MichaelOrrison,HarveyMuddCollege Lisette G. de Pillis, HarveyMuddCollege JillPipher,BrownUniversity FadilSantosa,UniversityofMinnesota UndergraduateTextsinMathematicsaregenerallyaimedatthird-andfourth-yearunder- graduatemathematicsstudentsatNorthAmericanuniversities.Thesetextsstrivetoprovide studentsandteacherswithnewperspectivesandnovelapproaches. Thebooksincludemo- tivation that guidesthe reader to an appreciationof interrelationsamongdifferent aspects ofthesubject. Theyfeatureexamplesthatillustrate keyconceptsaswellasexercisesthat strengthenunderstanding. Moreinformationaboutthisseriesathttp://www.springer.com/series/666 Charles C. Pugh Real Mathematical Analysis Second Edition CharlesC.Pugh DepartmentofMathematics UniversityofCalifornia Berkeley,CA,USA ISSN0172-6056 ISSN2197-5604 (electronic) UndergraduateTextsinMathematics ISBN978-3-319-17770-0 ISBN978-3-319-17771-7 (eBook) DOI10.1007/978-3-319-17771-7 LibraryofCongressControlNumber:2015940438 MathematicsSubjectClassification(2010):26-xx SpringerChamHeidelbergNewYorkDordrechtLondon © Springer International Publishing Switzerland 2002, 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) To Candida and to the students who have encouraged me – – especially A.W., D.H., and M.B. Preface Was plane geometry your favorite math course in high school? Did you like prov- ing theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is pure mathematics, and I hope it appeals to you, the budding pure mathematician. This book is set out for college juniors and seniors who love math and who profit from pictures that illustrate the math. Rarely is a picture a proof, but I hope a good picture will cement your understanding of why something is true. Seeing is believing. Chapter 1 gets you off the ground. The whole of analysis is built on the system of real numbers R, and especially on its Least Upper Bound property. Unlike many analysis texts that assume R and its properties as axioms, Chapter 1 contains a natural construction of R and a natural proof of the LUB property. You will also see why some infinite sets are more infinite than others, and how to visualize things in four dimensions. Chapter 2 is about metric spaces, especially subsets of the plane. This chapter contains many pictures you have never seen. (cid:2) and δ will become your friends. Most of the presentation uses sequences and limits, in contrast to open coverings. It may be less elegant but it’s easier to begin with. You will get to know the Cantor set well. Chapter 3 is about Freshman Calculus – differentiation, integration, L’Hˆopital’s Rule, and so on, for functions of a single variable – but this time you will find out why what you were taught before is actually true. In particular you will see that a bounded function is integrable if and only if it is continuous almost everywhere, and how this fact explains many other things about integrals. Chapter 4 is about functions viewed en masse. You can treat a set of functions as a metric space. The “points” in the space aren’t numbers or vectors – they are functions. What is the distance between two functions? What should it mean that a sequenceoffunctionsconvergestoalimitfunction? Whathappenstoderivativesand integrals when your sequence of functions converges to a limit function? When can you approximate a bad function with a good one? What is the best kind of function? What does the typical continuous function look like? (Answer: “horrible.”) Chapter 5 is about Sophomore Calculus – functions of several variables, partial derivatives, multiple integrals, and so on. Again you will see why what you were taught before is actually true. You will revisit Lagrange multipliers (with a picture vii viii Preface proof), the Implicit Function Theorem, etc. The main new topic for you will be differentialforms. Theyarepresentednotasmysterious“multi-indexedexpressions,” but rather as things that assign numbers to smooth domains. A 1-form assigns to a smooth curve a number, a 2-form assigns to a surface a number, a 3-form assigns to a solid a number, and so on. Orientation (clockwise, counterclockwise, etc.) is important and lets you see why cowlicks are inevitable – the Hairy Ball Theorem. The culmination of the differential forms business is Stokes’ Formula, which unifies what you know about div, grad, and curl. It also leads to a short and simple proof of the Brouwer Fixed Point Theorem – a fact usually considered too advanced for undergraduates. Chapter 6 is about Lebesgue measure and integration. It is not about measure theory in the abstract, but rather about measure theory in the plane, where you can see it. Surely I am not the first person to have rediscovered J.C. Burkill’s approach to the Lebesgue integral, but I hope you will come to value it as much as I do. After you understand a few nontrivial things about area in the plane, you are naturally led to define the integral as the area under the curve – the elementary picture you saw in high school calculus. Then the basic theorems of Lebesgue integration simply fall out from the picture. Included in the chapter is the subject of density points – points at which a set “clumps together.” I consider density points central to Lebesgue measure theory. At the end of each chapter are a great many exercises. Intentionally, there is no solution manual. You should expect to be confused and frustrated when you first try to solve the harder problems. Frustration is a good thing. It will strengthen you and it is the natural mental state of most mathematicians most of the time. Join the club! When you do solve a hard problem yourself or with a group of your friends, you will treasure it far more than something you pick up off the web. For encouragement, read Sam Young’s story at http://legacyrlmoore.org/reference/young.html. I have adopted Moe Hirsch’s star system for the exercises. One star is hard, two stars is very hard, and a three-star exercise is a question to which I do not know the answer. Likewise, starred sections are more challenging. Berkeley, California, USA Charles Chapman Pugh Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Real Numbers 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5* Comparing Cardinalities . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6* The Skeleton of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7* Visualizing the Fourth Dimension . . . . . . . . . . . . . . . . . . . . . 41 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2 A Taste of Topology 1 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3 The Topology of a Metric Space. . . . . . . . . . . . . . . . . . . . . . 65 4 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6 Other Metric Space Concepts . . . . . . . . . . . . . . . . . . . . . . . 92 7 Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8 Cantor Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 9* Cantor Set Lore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 10* Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 ix