An Introduction to Analysis (Global Edition)

696 Pages·2021·6.98 MB·English

by  William R. Wade

This is a special edition of an established title widely used by
GLOBAL GLOBAL
colleges and universities throughout the world. Pearson published
this exclusive edition for the benefit of students outside the
EDITION EDITION
United States and Canada. If you purchased this book within EG
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the United States or Canada, you should be aware that it has ITO
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been imported without the approval of the Publisher or Author. NAL

An Introduction to Analysis, now in its fourth edition, introduces the concepts of analysis in a
one-dimensional setting before delving deep into multidimensional theory. Building on students’
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existing knowledge of calculus and linear algebra, this text is a great resource for students
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taking courses in advanced calculus, Analysis I, or real analysis. The author makes the study

T I
of mathematics easier by beginning each abstruse topic with a snippet that describes, in a N
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nutshell, the concept about to be covered. With a collection of questions ranging from basic T

A R
to challenging, end-of-section exercises meet each student at their level, bringing beginners up
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to speed and offering additional practice to more advanced students. With its core material
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(which focuses on fundamental principles of analysis) and enrichment material (which explores
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challenging concepts), this text enables students to apply theory to practice and construct their Y
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own proofs. I T
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Key Features
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• Introductory material, previously scattered across several sections, has now been consolidated
and presented under a new section named ‘Introduction’ in Chapter 1.
• Assuming that students taking this course are familiar with basic calculus, this text has replaced
most of the beginning, computational, calculus-style exercises with slightly more conceptual
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ones, helping students develop a deeper understanding of the subject. D O
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• True–false questions scattered across the first six chapters clarify common misconceptions, and
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by splitting difficult questions into several parts, the book helps students get to solutions in a
N H AN INTRODUCTION
structured, step-by-step format.
• Complicated proofs are preceded by short paragraphs labeled ‘Strategy’, which demonstrate
why the proof works. TO ANALYSIS
• In addition to the core material it contains, the book also features enrichment sections and optional
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content, which instructors can also choose to assign to their students.
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FOURTH EDITION
WILLIAM R. WADE
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An Introduction to
Analysis
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An Introduction to
Analysis
Fourth Edition
Global Edition
WilliamR.Wade
UniversityofTennessee
Pearson Education Limited
KAO Two
KAO Park
Hockham Way
Harlow
CM17 9SR
United Kingdom
and Associated Companies throughout the world
Visit us on the World Wide Web at: www.pearsonglobaleditions.com
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© Pearson Education Limited 2022
The rights of William R. Wade to be identified as the author of this work have been asserted by them in accordance with the
Copyright, Designs and Patents Act 1988.
Authorized adaptation from the United States edition, entitled An Introduction to Analysis, 4th Edition, ISBN 978-0-13-
470762-4 by William R. Wade, published by Pearson Education © 2018.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or
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ISBN 10: 1-292-35787-8
ISBN 13: 978-1-292-35787-4
eBook ISBN 13: 978-1-292-35788-1
British Library Cataloguing-in-Publication Data
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Contents
Preface 10
PARTI ONE-DIMENSIONALTHEORY
1 TheRealNumberSystem 15
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 OrderedFieldAxioms . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 CompletenessAxiom . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4 MathematicalInduction . . . . . . . . . . . . . . . . . . . . . . . 37
1.5 InverseFunctionsandImages . . . . . . . . . . . . . . . . . . . 43
1.6 CountableandUncountableSets . . . . . . . . . . . . . . . . . . 49
2 SequencesinR 55
2.1 LimitsofSequences . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2 LimitTheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.3 Bolzano–WeierstrassTheorem . . . . . . . . . . . . . . . . . . . 67
2.4 CauchySequences . . . . . . . . . . . . . . . . . . . . . . . . . . 72
∗2.5 LimitsSupremumandInfimum . . . . . . . . . . . . . . . . . . . 75
3 FunctionsonR 82
3.1 Two-SidedLimits . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2 One-SidedLimitsandLimitsatInfinity . . . . . . . . . . . . . . 90
3.3 Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.4 UniformContinuity . . . . . . . . . . . . . . . . . . . . . . . . . 106
4 DifferentiabilityonR 112
4.1 TheDerivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2 DifferentiabilityTheorems . . . . . . . . . . . . . . . . . . . . . 119
4.3 TheMeanValueTheorem . . . . . . . . . . . . . . . . . . . . . . 123
4.4 Taylor’sTheoremandl’Hôpital’sRule . . . . . . . . . . . . . . . 131
4.5 InverseFunctionTheorems . . . . . . . . . . . . . . . . . . . . . 139
5 IntegrabilityonR 144
5.1 TheRiemannIntegral . . . . . . . . . . . . . . . . . . . . . . . . 144
5.2 RiemannSums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.3 TheFundamentalTheoremofCalculus . . . . . . . . . . . . . . 166
5.4 ImproperRiemannIntegration . . . . . . . . . . . . . . . . . . . 177
∗5.5 FunctionsofBoundedVariation . . . . . . . . . . . . . . . . . . 184
∗5.6 ConvexFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 189
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8 Contents
6 InfiniteSeriesofRealNumbers 198
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
6.2 SerieswithNonnegativeTerms . . . . . . . . . . . . . . . . . . . 206
6.3 AbsoluteConvergence . . . . . . . . . . . . . . . . . . . . . . . 212
6.4 AlternatingSeries . . . . . . . . . . . . . . . . . . . . . . . . . . 223
∗6.5 EstimationofSeries . . . . . . . . . . . . . . . . . . . . . . . . . 228
∗6.6 AdditionalTests . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
7 InfiniteSeriesofFunctions 236
7.1 UniformConvergenceofSequences . . . . . . . . . . . . . . . . 236
7.2 UniformConvergenceofSeries . . . . . . . . . . . . . . . . . . 244
7.3 PowerSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
7.4 AnalyticFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 263
∗7.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
PARTII MULTIDIMENSIONALTHEORY
8 EuclideanSpaces 281
8.1 AlgebraicStructure . . . . . . . . . . . . . . . . . . . . . . . . . 281
8.2 PlanesandLinearTransformations. . . . . . . . . . . . . . . . . 293
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9 ConvergenceinR 302
9.1 TopologyofRn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
9.2 Interior,Closure,andBoundary . . . . . . . . . . . . . . . . . . 311
∗9.3 CompactSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
9.4 Heine–BorelTheorem . . . . . . . . . . . . . . . . . . . . . . . . 319
9.5 LimitsofSequences . . . . . . . . . . . . . . . . . . . . . . . . . 324
9.6 LimitsofFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . 328
9.7 ContinuousFunctions . . . . . . . . . . . . . . . . . . . . . . . . 338
∗9.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
10 MetricSpaces 356
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
10.2 Interior,Closure,andBoundary . . . . . . . . . . . . . . . . . . 364
10.3 CompactSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
10.4 ConnectedSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
10.5 LimitsofFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . 382
10.6 ContinuousFunctions . . . . . . . . . . . . . . . . . . . . . . . . 386
∗10.7 Stone–WeierstrassTheorem. . . . . . . . . . . . . . . . . . . . . 391
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11 DifferentiabilityonR 397
11.1 PartialDerivativesandPartialIntegrals . . . . . . . . . . . . . . 397
11.2 TheDefinitionofDifferentiability . . . . . . . . . . . . . . . . . 408
11.3 Derivatives,Differentials,andTangentPlanes . . . . . . . . . . 417
11.4 TheChainRule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
11.5 TheMeanValueTheoremandTaylor’sFormula . . . . . . . . . 430
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Contents 9
11.6 TheInverseFunctionTheorem . . . . . . . . . . . . . . . . . . . 438
∗11.7 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
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12 IntegrationonR 463
12.1 JordanRegions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
12.2 RiemannIntegrationonJordanRegions . . . . . . . . . . . . . 476
12.3 IteratedIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
12.4 ChangeofVariables . . . . . . . . . . . . . . . . . . . . . . . . . 504
∗12.5 PartitionsofUnity . . . . . . . . . . . . . . . . . . . . . . . . . . 517
∗12.6 TheGammaFunctionandVolume . . . . . . . . . . . . . . . . . 528
13 FundamentalTheoremsofVectorCalculus 537
13.1 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
13.2 OrientedCurves . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
13.3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
13.4 OrientedSurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 569
13.5 TheoremsofGreenandGauss . . . . . . . . . . . . . . . . . . . 579
13.6 Stokes’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . 589
14 FourierSeries 598
∗14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
∗14.2 SummabilityofFourierSeries . . . . . . . . . . . . . . . . . . . 605
∗14.3 GrowthofFourierCoefficients . . . . . . . . . . . . . . . . . . . 612
∗14.4 ConvergenceofFourierSeries . . . . . . . . . . . . . . . . . . . 620
∗14.5 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
Appendices 633
A. AlgebraicLaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
B. Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
C. MatricesandDeterminants . . . . . . . . . . . . . . . . . . . . . 643
D. QuadricSurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 651
E. VectorCalculusandPhysics . . . . . . . . . . . . . . . . . . . . . 655
F. EquivalenceRelations . . . . . . . . . . . . . . . . . . . . . . . . 658
References 660
AnswersandHintstoSelectedExercises 661
SubjectIndex 681
NotationIndex 693