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The Real Numbers and Real Analysis Ethan D. Bloch The Real Numbers and Real Analysis Ethan D. Bloch Mathematics Department Bard College Annandale-on-Hudson, NY 12504 USA bloch@bard.edu ISBN 978-0-387-72176-7 e-ISBN 978-0-387-72177-4 DOI 10.1007/978-0-387-72177-4 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011928556 Mathematics Subject Classification (2010): 26-01 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) DedicatedtomytwowonderfulchildrenGilNehemyaandAdaHaviva, forwhommylovehasnoupperbound Contents Preface............................................................ xi TotheStudent ..................................................... xxi TotheInstructor ...................................................xxvii 1 ConstructionoftheRealNumbers ................................ 1 1.1 Introduction ................................................ 1 1.2 Entry1:AxiomsfortheNaturalNumbers ...................... 2 1.3 ConstructingtheIntegers ..................................... 11 1.4 Entry2:AxiomsfortheIntegers .............................. 19 1.5 ConstructingtheRationalNumbers ........................... 27 1.6 DedekindCuts............................................. 33 1.7 ConstructingtheRealNumbers................................ 41 1.8 HistoricalRemarks .......................................... 51 2 PropertiesoftheRealNumbers................................... 61 2.1 Introduction ................................................ 61 2.2 Entry3:AxiomsfortheRealNumbers......................... 62 2.3 AlgebraicPropertiesoftheRealNumbers ...................... 65 2.4 Finding the Natural Numbers, the Integers and the Rational NumbersintheRealNumbers................................ 75 2.5 InductionandRecursioninPractice ........................... 83 2.6 TheLeastUpperBoundPropertyandItsConsequences .......... 96 2.7 UniquenessoftheRealNumbers.............................. 107 2.8 DecimalExpansionofRealNumbers.......................... 113 2.9 HistoricalRemarks ......................................... 128 3 LimitsandContinuity .......................................... 129 3.1 Introduction ............................................... 129 3.2 LimitsofFunctions......................................... 129 viii Contents 3.3 Continuity ................................................ 146 3.4 UniformContinuity......................................... 156 3.5 TwoImportantTheorems .................................... 163 3.6 HistoricalRemarks ..........................................171 4 Differentiation..................................................181 4.1 Introduction ................................................181 4.2 TheDerivative ..............................................181 4.3 ComputingDerivatives...................................... 192 4.4 TheMeanValueTheorem ................................... 198 4.5 IncreasingandDecreasingFunctions,PartI:LocalandGlobal Extrema .................................................. 207 4.6 IncreasingandDecreasingFunctions,PartII:FurtherTopics ...... 215 4.7 HistoricalRemarks ......................................... 225 5 Integration .....................................................231 5.1 Introduction ................................................231 5.2 TheRiemannIntegral ........................................231 5.3 ElementaryPropertiesoftheRiemannIntegral .................. 242 5.4 UpperSumsandLowerSums ................................ 247 5.5 FurtherPropertiesoftheRiemannIntegral ..................... 258 5.6 FundamentalTheoremofCalculus ............................ 267 5.7 ComputingAntiderivatives................................... 277 5.8 Lebesgue’sTheorem........................................ 283 5.9 AreaandArcLength........................................ 293 5.10 HistoricalRemarks ......................................... 312 6 LimitstoInfinity................................................321 6.1 Introduction ................................................321 6.2 LimitstoInfinity ........................................... 322 6.3 ComputingLimitstoInfinity ..................................331 6.4 ImproperIntegrals...........................................341 6.5 HistoricalRemarks ......................................... 354 7 TranscendentalFunctions....................................... 357 7.1 Introduction ............................................... 357 7.2 LogarithmicandExponentialFunctions........................ 358 7.3 TrigonometricFunctions .................................... 369 7.4 Moreaboutπ .............................................. 379 7.5 HistoricalRemarks ..........................................391 8 Sequences ..................................................... 399 8.1 Introduction ............................................... 399 8.2 Sequences................................................. 399 8.3 ThreeImportantTheorems................................... 412 8.4 ApplicationsofSequences ................................... 423 Contents ix 8.5 HistoricalRemarks ......................................... 439 9 Series......................................................... 443 9.1 Introduction ............................................... 443 9.2 Series .................................................... 443 9.3 ConvergenceTests...........................................451 9.4 AbsoluteConvergenceandConditionalConvergence............. 459 9.5 PowerSeriesasFunctions ................................... 473 9.6 HistoricalRemarks ......................................... 482 10 SequencesandSeriesofFunctions ............................... 489 10.1 Introduction ............................................... 489 10.2 SequencesofFunctions ..................................... 489 10.3 SeriesofFunctions ......................................... 502 10.4 FunctionsasPowerSeries ................................... 509 10.5 AContinuousbutNowhereDifferentiableFunction.............. 527 10.6 HistoricalRemarks ......................................... 534 Bibliography....................................................... 539 Index ............................................................. 545

Description:

This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. The choice of material and the flexible organization, including three different entryways into the study of the real numbers,