AnalysisITerenceTaothirdedition

第三版The Texts and Readings in Mathematics series publishes high-quality textbooksresearch-level monographs, lecture notes and contributed volumes. Undergraduateand graduate students of mathematics, research scholars, and teachers would findthis book series useful. The volumes are carefully written as teaching aids andhighlight characteristic features of the theory. The books in this series areco-published with Hindustan Book Agency, New Delhi, IndiaMoreinformationaboutthisseriesathttp://www.springer.com/series/15141Terence taoAnalySiSThird edition应 HINDUSTAN囗几 BOOK AGENCY②SpringerTerence taoDepartment of mathematicsUniversity of California, Los AngelesLoS Angeles, CAUSAThis work is a co-publication with hindustan book agency, New Delhi, licensed for sale in allcountries in electronic form only. Sold and distributed in print across the world by HindustanBook agency, P-19 Green Park Extension, New Delhi 110016, India. ISBN: 978-93-80250-64-9C Hindustan book agency 2015issn 2366-8725(electronic)Texts and readings in mathematicsISBN978981-10-1789-6( e BookDOI10.1007/978-981-10-1789-6Library of Congress Control Number: 2016940817C Springer Science+Business Media Singapore 2016 and Hindustan Book Agency 2015This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or partof 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 transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developedThe use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general useThe publishers, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publishers nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been madeThis Springer imprint is published by Springer NatureThe registered company is Springer Science+Business Media Singapore Pte LtdTo my parents, for everythingContentsPreface to the second and third editionsPreface to the first editionabout the authorIntroduction1. 1 What is analysis?1.2 Why do analysis?2 Starting at the beginning: the natural numbers132.1 The peano axioms2.2 Addition242. 3 Multiplication293 Set theory333.1 Fundamentals333.2 Russells paradox(Optional)3.3 Functions493.4 Images and inverse images563.5 Cartesian products623.6 Cardinality of sets674 Integers and rationals744.1 The integers744.2 The rationals4.3 Absolute value and exponentiation864.4 Gaps in the rational numbers5 The real numbers945.1 Cauchy sequences5.2 Equivalent Cauchy sequences5.3 The construction of the real numbers1025.4 Ordering the reals111Contents5.5 The least upper bound property1165.6 Real exponentiation, part I1216 Limits of sequences1266.1 Convergence and limit laws1266.2 The Extended real number system1336.3 Suprema and Infima of sequences1376.4 Limsup, Liminf, and limit points1396.5 Some standard limits1486.6 Subsequences1496.7 Real exponentiation, part II1527 Series1557.1 Finite series1557.2 Infinite series1647.3 Sums of non-negative numbers.1707.4 Rearrangement of series1747. 5 The root and ratio tests.1788 Infinite sets1818.1 Countability1818.2ummation on infinite sets1888.3 Uncountable sets1958.4 The axiom of choice1988.5 Ordered sets2029 Continuous functions on R2119.1 Subsets of the real line2119.2 The algebra of real-valued functions2179.3 Limiting values of functions.2209.4 Continuous functions2279.5 Left and right limits2319.6 The maximum principle2349. 7 The intermediate value theorem2389. 8 Monotonic functions2419.9 Uniform continuity.2439.10 Limits at infinit24910 Differentiation of functions25110.1 Basic definitions251Contents10.2 Local maxima. local minima and derivatives.25710.3 Monotone functions and derivatives26010.4 Inverse functions and derivatives26110.5 L'HOpital's rule26411 The Riemann integral26711.1 Partitions2611.2 Piecewise constant functions.27211.3 Upper and lower Riemann integrals27611.4 Basic properties of the riemann integr28011.5 Riemann integrability of continuous functions28511.6 Riemann integrability of monotone functions28911.7 A non-Riemann integrable function29111. 8 The Riemann-Stieltjes integral29211.9 The two fundamental theorems of calculus29511.10 Consequences of the fundamental theorems300A Appendix: the basics of mathematical logic305A 1 Mathematical statements306A2 Implication312A. 3 The structure of proofs.317A 4 Variables and quantifiers320A.5 Nested quantifiers324A. 6 Some examples of proofs and quantifiers327A 7 Equality.329B Appendix: the decimal system331B. 1 The decimal representation of natural numbers332B. 2 The decimal representation of real numbers335Index339Texts and Readings in Mathematics349Preface to the second and third editionsSince the publication of the first edition, many students and lecturers have communicated a number of minor typos and other correctionsto me. There was also some demand for a hardcover edition of thetexts. Because of this, the publishers and i have decided to incorporatethe corrections and issue a hardcover second edition of the textbooksThe layout, page numbering, and indexing of the texts have also beenchanged; in particular the two volumes are now numbered and indexedseparately. However, the chapter and exercise numbering, as well as themathematical content. remains the same as the first edition, and so thetwo editions can be used more or less interchangeably for homework andstudy purposesThe third edition contains a number of corrections that were reportedfor the second edition, together with a few new exercises, but is otherwiseessentially the same text