Elementary Analysis - People

1 Undergraduate Texts in MathematicsElementary AnalysisKenneth A. RossThe Theory of CalculusSecond EditionUndergraduate Texts in MathematicsUndergraduate Texts in MathematicsSeries Editors:Sheldon AxlerSan Francisco State University, San Francisco, CA, USAK enneth RibetUniversity of California, Berkeley, CA, USAA dvisory Board:Colin C. Adams,Williams College, Williamstown, MA, USAA lejandro Adem,University of British Columbia, Vancouver, BC, CanadaRuth Charney,Brandeis University, Waltham, MA, USAI rene M. Gamba,The University of Texas at Austin, Austin, TX, USAR oger E. Howe,Yale University, New Haven, CT, USAD avid Jerison,Massachusetts Institute of Technology, Cambridge, MA, USAJ effrey C. Lagarias,University of Michigan, Ann Arbor, MI, USAJill Pipher,Brown University, Providence, RI, USAF adil Santosa,University of Minnesota, Minneapolis, MN, USAAmie Wilkinson,University of Chicago, Chicago, IL, USAU ndergraduate Texts in Mathematicsare generally aimed at third- and fourth-year undergraduate mathematics students at North American universities.

2 These textsstrive to provide students and teachers with new perspectives and novel books include motivation that guides the reader to an appreciation of interrela-tions among different aspects of the subject. They feature examples that illustrate keyconcepts as well as exercises that strengthen further volumes: A. RossElementary AnalysisThe Theory of CalculusSecond EditionIn collaboration with Jorge M. L opez, University ofPuerto Rico, R o Piedras123 Kenneth A. RossDepartment of MathematicsUniversity of OregonEugene, OR, USAISSN 0172-6056 ISBN 978-1-4614-6270-5 ISBN 978-1-4614-6271-2 (eBook)DOI New York Heidelberg Dordrecht LondonLibrary of Congress Control Number: 2013950414 Mathematics Subject Classification: 26-01, 00-01, 26A06, 26A24, 26A27, 26A42 Springer Science+Business Media New York2013 This work is subject to copyright.

3 All rights are reserved by the Publisher, 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 transmission orinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissim-ilar methodology now known or hereafter developed. Exempted from this legal reservation are briefexcerpts in connection with reviews or scholarly Analysis or material supplied specifically for the pur-pose of being entered and executed on a computer system, for exclusive use by the purchaser of thework. Duplication of this publication or parts thereof is permitted onlyunder the provisions of theCopyright Law of the Publisher s location, in its current version, and permission for use must alwaysbe obtained from Springer.

4 Permissions for use may be obtained through RightsLink at the CopyrightClearance Center. Violations are liable to prosecution under the respective Copyright use of general descriptive names, registered names, trademarks, service marks, etc. in this publi-cation 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 the advice and information in this book are believed to be true andaccurate at the date ofpublication, neither the authors nor the editors nor the publisher can accept any legal responsibility forany errors or omissions that may be made. The publisher makes no warranty, express or implied, withrespect to the material contained on acid-free paperSpringer is part of Springer Science+Business Media ( )PrefacePreface to the First EditionA study of this book, and espe-cially the exercises, should give the reader a thorough understandingof a few basic concepts in Analysis such as continuity, convergenceof sequences and series of numbers, and convergence of sequencesand series of functions.

5 An ability to read and write proofs willbe stressed. A precise knowledge of definitions is essential. The be-ginner should memorize them; such memorization will help lead the scene and, except for the completeness axiom,should be more or less familiar. Accordingly, readers and instructorsare urged to move quickly through this chapter and refer back to itwhen necessary. The most critical sections in the book are 7 If these sections are thoroughly digested and understood,the remainder of the book should be smooth first four chapters form a unit for a short course on cover these four chapters (except for the enrichment sections and 20) in about 38 class periods; this includes time for quizzes andexaminations. For such a short course, my philosophy is that thestudents are relatively comfortable with derivatives and integrals butdo not really understand sequences and series, much less sequencesand series of functions, so 4focus on these topics.

6 On twovPrefacevior three occasions, I draw on the Fundamental Theorem of Calculusor the Mean Value Theorem, which appears later in the book, but ofcourse these important theorems are at least discussed in a standardcalculus the early sections, especially in , the proofs are verydetailed with careful references for even the most Elementary sophisticated readers find excessive details and references ahindrance (they break the flow of the proof and tend to obscure themain ideas) and would prefer to check the items mentally as theyproceed. Accordingly, in later chapters, the proofs will be somewhatless detailed, and references for the simplest facts will often be omit-ted. This should help prepare the reader for more advanced bookswhich frequently give very brief of the basic conceptsin this book should make theanalysis in such areas as complex variables, differential equations,numerical Analysis , and statistics more meaningful.

7 The book canalso serve as a foundation for an in-depth study of real analysisgiveninbookssuchas[4,33,34,53,62 ,65] listed in the planning to teach calculus will also benefit from a carefulstudy of Analysis . Even after studying this book (or writing it), it willnot be easy to handle questions such as What is a number? butat least this book should help give a clearer picture of the subtletiesto which such questions enrichment sections contain discussions of some topics that Ithink are important or interesting. Sometimes the topic is dealt withlightly, and suggestions for further reading are given. Though thesesections are not particularly designed for classroom use, I hope thatsome readers will use them to broaden their horizons and see howthis material fits in the general scheme of have benefitted from numerous helpful suggestions from my col-leagues Robert Freeman, William Kantor, Richard Koch, and JohnLeahy and from Timothy Hall, Gimli Khazad, and Jorge L opez.

8 Ihave also had helpful conversations with my wife Lynn concerninggrammar and taste. Of course, remaining errors in grammar andmathematics are the responsibility of the users have supplied me with corrections and suggestionsthat I ve incorporated in subsequent printings. I thank them all,Prefaceviiincluding Robert Messer of Albion College, who caught a subtle errorin the proof of to the Second EditionAfter 32 years, it seemed timeto revise this book. Since the first edition was so successful, I haveretained the format and material from the first edition. The num-bering of theorems, examples, and exercises in each section will bethe same, and new material will be added to some of the rule has an exception, and this rule is no exception. In 11,a theorem ( ) has been added, which allows the sim-plification of four almost-identical proofs in the section: Examples3and4, (formerly Corollary ), and (formerly Theorem ).

9 Where appropriate, the presentation has been improved. See es-pecially the proof of the Chain , the shorter proof of Abel , and the shorter treatment of decimal expansions in 16. Also, a few examples have been added, a few exercises have beenmodified or added, and a couple of exercises have been are the main additions to this revision. The proof of theirrationality ofein 16is now accompanied by an elegant proof that is also irrational. Even though this is an enrichment section,it is especially recommended for those who teach or will teach pre-college mathematics. The Baire Category Theorem and interestingconsequences have been added to the enrichment 21. Section31,onTaylor s Theorem, has been overhauled. It now includes a discussionof Newton s method for approximating zeros of functions, as wellas its cousin, the secant method.

10 Proofs are provided for theoremsthat guarantee when these approximation methods work. Section35on Riemann-Stieltjes integrals has been improved and new section, 38, contains an example of a continuous nowhere-differentiable function and a theorem that shows most continuousfunctions are nowhere differentiable. Also, each of 22,32,and33has been modestly is a pleasure to thank many People who have helped overthe years since the first edition appeared in 1980. This includesDavid M. Bloom, Robert B. Burckel, Kai Lai Chung, Mark Dalthorp(grandson), M. K. Das (India), Richard Dowds, Ray Hoobler,PrefaceviiiRichard M. Koch, Lisa J. Madsen, Pablo V. Negr on Marrero(Puerto Rico), Rajiv Monsurate (India), Theodore W. Palmer, J urgR atz (Switzerland), Peter Renz, Karl Stromberg, and Jes us Sueiras(Puerto Rico).