Solutions Manual for An Introduction to Analysis (Classic) 4th Ed – Wade 2018

Covers All 14 Chapters
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SOLUTIONS TO EXERCISES
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, An Introduction to Analysis
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Table of Contents dr dr




Chapter 1: The Real Number System
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1.2 Ordered field axioms................................................................. 1
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1.3 The Completeness Axiom…...................................................... 2
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1.4 Mathematical Induction… ......................................................... 4
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1.5 Inverse Functions and Images… ............................................... 6
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1.6 Countable and uncountable sets…............................................. 8
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Chapter 2: Sequences in R
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2.1 Limits of Sequences… ............................................................. 10
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2.2 Limit Theorems........................................................................ 11
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2.3 Bolzano-Weierstrass Theorem .................................................. 13 dr


2.4 Cauchy Sequences… ................................................................ 15
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2.5 Limits Supremum and Infimum................................................ 16
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Chapter 3: Functions on R
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3.1 Two-Sided Limits…................................................................. 19
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3.2 One-Sided Limits and Limits at Infinity… ................................20
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3.3 Continuity…............................................................................. 22
3.4 Uniform Continuity… .............................................................. 24
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Chapter 4: Differentiability on R
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4.1 The Derivative…...................................................................... 27
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4.2 Differentiability Theorem… ..................................................... 28
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4.3 The Mean Value Theorem….................................................... 30
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4.4 Taylor’s Theorem and l’Hôpital’s Rule… ................................ 32
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4.5 Inverse Function Theorems ...................................................... 34
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Chapter 5: Integrability on R
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5.1 The Riemann Integral… ............................................................ 37
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5.2 Riemann Sums .......................................................................... 40
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5.3 The Fundamental Theorem of Calculus… ................................. 43
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5.4 Improper Riemann Integration… ............................................... 46
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5.5 Functions of Bounded Variation… ............................................ 49
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5.6 Convex Functions… ................................................................. 51
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Copyright © 2010 Pearson Education, Inc. dr dr dr dr dr d r Publishing as Prentice
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,Chapter 6: Infinite Series of Real Numbers
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6.1 Introduction… ........................................................................... 53
6.2 Series with Nonnegative Terms… ............................................. 55
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6.3 Absolute Convergence… ........................................................... 57
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6.4 Alternating Series… .................................................................. 60
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6.5 Estimation of Series… .............................................................. 62
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6.6 Additional Tests… .................................................................... 63
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Chapter 7: Infinite Series of Functions
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7.1 Uniform Convergence of Sequences… ...................................... 65
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7.2 Uniform Convergence of Series… ............................................ 67
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7.3 Power Series… ......................................................................... 69
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7.4 Analytic Functions…................................................................ 72
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7.5 Applications….......................................................................... 74

Chapter 8: Euclidean Spaces
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8.1 Algebraic Structure…............................................................... 76
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8.2 Planes and Linear Transformations… ....................................... 77
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8.3 Topology of Rn ............................................................................................................ 79
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8.4 Interior, Closure, and Boundary… ........................................... 80
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Chapter 9: Convergence in Rn
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9.1 Limits of Sequences… ............................................................. 82
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9.2 Heine-Borel Theorem ................................................................ 83
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9.3 Limits of Functions… ............................................................... 84
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9.4 Continuous Functions… ............................................................. 86
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9.5 Compact Sets… ........................................................................ 87
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9.6 Applications…........................................................................... 88

Chapter 10: Metric Spaces
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10.1 Introduction… ............................................................................ 90
10.2 Limits of Functions… ................................................................ 91
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10.3 Interior, Closure, and Boundary… ............................................. 92
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10.4 Compact Sets… ......................................................................... 93
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10.5 Connected Sets… ...................................................................... 94
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10.6 Continuous Functions… .............................................................. 96
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10.7 Stone-Weierstrass Theorem ........................................................ 97
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Copyright © 2010 Pearson Education, Inc.
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Hall.
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, Chapter 11: Differentiability on Rn
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11.1 Partial Derivatives and Partial Integrals… .................................... 99
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11.2 The Definition of Differentiability… ............................................102
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11.3 Derivatives, Differentials, and Tangent Planes… .........................104
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11.4 The Chain Rule… ........................................................................ 107
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11.5 The Mean Value Theorem and Taylor’s Formula…..................... 108
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11.6 The Inverse Function Theorem..................................................... 111
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11.7 Optimization… ..............................................................................114

Chapter 12: Integration on Rn
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12.1 Jordan Regions… .......................................................................... 117
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12.2 Riemann Integration on Jordan Regions… .................................... 119
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12.3 Iterated Integrals… ......................................................................... 122
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12.4 Change of Variables… .................................................................. 125
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12.5 Partitions of Unity… ..................................................................... 130
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12.6 The Gamma Function and Volume................................................ 131
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Chapter 13: Fundamental Theorems of Vector Calculus
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13.1 Curves… ........................................................................................ 135
13.2 Oriented Curves… ......................................................................... 137
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13.3 Surfaces…...................................................................................... 140
13.4 Oriented Surfaces… ....................................................................... 143
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13.5 Theorems of Green and Gauss… ................................................... 147
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13.6 Stokes’s Theorem........................................................................... 150
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Chapter 14: Fourier Series
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14.1 Introduction… ................................................................................ 156
14.2 Summability of Fourier Series… .................................................... 157
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14.3 Growth of Fourier Coefficients….................................................. 159
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14.4 Convergence of Fourier Series… .................................................. 160
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14.5 Uniqueness… ................................................................................. 163




Copyright © 2010 Pearson Education, Inc.
dr dr dr dr dr d r Publishing as Prentice
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Hall.
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