Texts and Readings in Mathematics 37 Terence Tao Analysis I Third Edition Texts and Readings in Mathematics Volume 37 Advisory Editor C.S. Seshadri, Chennai Mathematical Institute, Chennai Managing Editor Rajendra Bhatia, Indian Statistical Institute, New Delhi Editor Manindra Agrawal, Indian Institute of Technology Kanpur, Kanpur V. Balaji, Chennai Mathematical Institute, Chennai R.B. Bapat, Indian Statistical Institute, New Delhi V.S. Borkar, Indian Institute of Technology Bombay, Mumbai T.R. Ramadas, Chennai Mathematical Institute, Chennai V. Srinivas, Tata Institute of Fundamental Research, Mumbai The Texts and Readings in Mathematics series publishes high-quality textbooks, research-level monographs, lecture notes and contributed volumes. Undergraduate and graduate students of mathematics, research scholars, and teachers would find this book series useful. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. The books in this series are co-published with Hindustan Book Agency, New Delhi, India. More information about this series at http://www.springer.com/series/15141 Terence Tao Analysis I Third Edition 123 Terence Tao Department of Mathematics University of California, Los Angeles Los Angeles, CA USA This work is a co-publication with Hindustan Book Agency, New Delhi, licensed for sale in all countries in electronic form only. Sold and distributed in print across the world by Hindustan Book Agency, P-19 Green Park Extension, New Delhi 110016, India. ISBN: 978-93-80250-64-9 © Hindustan Book Agency 2015. ISSN 2366-8725 (electronic) Texts and Readings in Mathematics ISBN 978-981-10-1789-6 (eBook) DOI 10.1007/978-981-10-1789-6 Library of Congress Control Number: 2016940817 © Springer Science+Business Media Singapore 2016 and Hindustan Book Agency 2015 This work is subject to copyright. 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To my parents, for everything Contents Preface to the second and third editions xi Preface to the first edition xiii About the Author xix 1 Introduction 1 1.1 Whatisanalysis?...................... 1 1.2 Whydoanalysis?...................... 2 2 Starting at the beginning: the natural numbers 13 2.1 ThePeanoaxioms...................... 15 2.2 Addition........................... 24 2.3 Multiplication........................ 29 3 Set theory 33 3.1 Fundamentals ........................ 33 3.2 Russell’s paradox (Optional) ................ 46 3.3 Functions........................... 49 3.4 Imagesandinverseimages................. 56 3.5 Cartesianproducts..................... 62 3.6 Cardinalityofsets...................... 67 4 Integers and rationals 74 4.1 Theintegers......................... 74 4.2 Therationals......................... 81 4.3 Absolutevalueandexponentiation............. 86 4.4 Gapsintherationalnumbers................ 90 5 The real numbers 94 5.1 Cauchysequences...................... 96 5.2 EquivalentCauchysequences................100 5.3 Theconstructionoftherealnumbers...........102 5.4 Orderingthereals......................111 vii viii Contents 5.5 Theleastupperboundproperty..............116 5.6 Realexponentiation,partI.................121 6 Limits of sequences 126 6.1 Convergenceandlimitlaws.................126 6.2 TheExtendedrealnumbersystem.............133 6.3 Suprema and Infima of sequences .............137 6.4 Limsup,Liminf,andlimitpoints..............139 6.5 Somestandardlimits....................148 6.6 Subsequences.........................149 6.7 Realexponentiation,partII................152 7 Series 155 7.1 Finiteseries.........................155 7.2 Infinite series .........................164 7.3 Sumsofnon-negativenumbers...............170 7.4 Rearrangementofseries...................174 7.5 Therootandratiotests...................178 8 Infinite sets 181 8.1 Countability .........................181 8.2 Summation on infinite sets .................188 8.3 Uncountablesets.......................195 8.4 Theaxiomofchoice.....................198 8.5 Orderedsets.........................202 9 Continuous functions on R 211 9.1 Subsetsoftherealline...................211 9.2 The algebra of real-valued functions ............217 9.3 Limiting values of functions ................220 9.4 Continuous functions ....................227 9.5 Leftandrightlimits.....................231 9.6 Themaximumprinciple...................234 9.7 Theintermediatevaluetheorem..............238 9.8 Monotonic functions .....................241 9.9 Uniformcontinuity.....................243 9.10 Limits at infinity .......................249 10 Differentiation of functions 251 10.1Basicdefinitions.......................251 Contents ix 10.2Localmaxima,localminima,andderivatives.......257 10.3 Monotone functions and derivatives ............260 10.4 Inverse functions and derivatives ..............261 10.5 L’Hˆopital’srule.......................264 11 The Riemann integral 267 11.1Partitions...........................268 11.2 Piecewise constant functions ................272 11.3UpperandlowerRiemannintegrals............276 11.4BasicpropertiesoftheRiemannintegral.........280 11.5 Riemann integrability of continuous functions ......285 11.6 Riemann integrability of monotone functions .......289 11.7 A non-Riemann integrable function ............291 11.8TheRiemann-Stieltjesintegral...............292 11.9 The two fundamental theorems of calculus ........295 11.10 Consequences of the fundamental theorems ........300 A Appendix: the basics of mathematical logic 305 A.1 Mathematicalstatements..................306 A.2 Implication..........................312 A.3 Thestructureofproofs...................317 A.4 Variablesandquantifiers..................320 A.5 Nestedquantifiers......................324 A.6 Someexamplesofproofsandquantifiers.........327 A.7 Equality...........................329 B Appendix: the decimal system 331 B.1 The decimal representation of natural numbers ......332 B.2 The decimal representation of real numbers ........335 Index 339 Texts and Readings in Mathematics 349 Preface to the second and third editions Since the publication of the first edition, many students and lectur- ers have communicated a number of minor typos and other corrections to me. There was also some demand for a hardcover edition of the texts. Because of this, the publishers and I have decided to incorporate the corrections and issue a hardcover second edition of the textbooks. The layout, page numbering, and indexing of the texts have also been changed; in particular the two volumes are now numbered and indexed separately. However, the chapter and exercise numbering, as well as the mathematical content, remains the same as the first edition, and so the two editions can be used more or less interchangeably for homework and study purposes. The third edition contains a number of corrections that were reported for the second edition, together with a few new exercises, but is otherwise essentially the same text. xi Preface to the first edition 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 arealnumber,or