Covers All 14 Chapters
SOLUTIONS TO EXERCISES
, An Introduction to Analysis
Table of Contents
Chapter 1: The Real Number System
1.2 Ordered field axioms ........................................................... 1
1.3 The Completeness Axiom… ................................................2
1.4 Mathematical Induction… .................................................... 4
1.5 Inverse Functions and Images… ......................................... 6
1.6 Countable and uncountable sets… .....................................8
Chapter 2: Sequences in R
2.1 Limits of Sequences…....................................................... 10
2.2 Limit Theorems...................................................................11
2.3 Bolzano-Weierstrass Theorem ...........................................13
2.4 Cauchy Sequences… ......................................................... 15
2.5 Limits Supremum and Infimum .......................................... 16
Chapter 3: Functions on R
3.1 Two-Sided Limits… ............................................................ 19
3.2 One-Sided Limits and Limits at Infinity… ............................ 20
3.3 Continuity… ........................................................................ 22
3.4 Uniform Continuity… ..........................................................24
Chapter 4: Differentiability on R
4.1 The Derivative… ................................................................. 27
4.2 Differentiability Theorem… ................................................. 28
4.3 The Mean Value Theorem… .............................................. 30
4.4 Taylor’s Theorem and l’Hôpital’s Rule… ........................... 32
4.5 Inverse Function Theorems ................................................ 34
Chapter 5: Integrability on R
5.1 The Riemann Integral… ...................................................... 37
5.2 Riemann Sums .................................................................... 40
5.3 The Fundamental Theorem of Calculus… ...........................43
5.4 Improper Riemann Integration… ........................................ 46
5.5 Functions of Bounded Variation… ...................................... 49
5.6 Convex Functions… ........................................................... 51
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
,Chapter 6: Infinite Series of Real Numbers
6.1 Introduction… .......................................................................53
6.2 Series with Nonnegative Terms… ....................................... 55
6.3 Absolute Convergence…..................................................... 57
6.4 Alternating Series…............................................................. 60
6.5 Estimation of Series… ......................................................... 62
6.6 Additional Tests… ............................................................... 63
Chapter 7: Infinite Series of Functions
7.1 Uniform Convergence of Sequences… ............................... 65
7.2 Uniform Convergence of Series… ....................................... 67
7.3 Power Series… ................................................................... 69
7.4 Analytic Functions… ............................................................72
7.5 Applications…..................................................................... 74
Chapter 8: Euclidean Spaces
8.1 Algebraic Structure… ......................................................... 76
8.2 Planes and Linear Transformations… ................................ 77
8.3 Topology of Rn.......................................................................................................... 79
8.4 Interior, Closure, and Boundary… ...................................... 80
Chapter 9: Convergence in Rn
9.1 Limits of Sequences….........................................................82
9.2 Heine-Borel Theorem .......................................................... 83
9.3 Limits of Functions… ........................................................... 84
9.4 Continuous Functions… ...................................................... 86
9.5 Compact Sets… .................................................................. 87
9.6 Applications…...................................................................... 88
Chapter 10: Metric Spaces
10.1 Introduction… ....................................................................... 90
10.2 Limits of Functions… ............................................................ 91
10.3 Interior, Closure, and Boundary… ........................................ 92
10.4 Compact Sets… ................................................................... 93
10.5 Connected Sets… ................................................................94
10.6 Continuous Functions… ....................................................... 96
10.7 Stone-Weierstrass Theorem................................................. 97
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
, Chapter 11: Differentiability on Rn
11.1 Partial Derivatives and Partial Integrals… .............................. 99
11.2 The Definition of Differentiability…..........................................102
11.3 Derivatives, Differentials, and Tangent Planes… ...................104
11.4 The Chain Rule… ................................................................... 107
11.5 The Mean Value Theorem and Taylor’s Formula…................ 108
11.6 The Inverse Function Theorem .............................................. 111
11.7 Optimization… .........................................................................114
Chapter 12: Integration on Rn
12.1 Jordan Regions….................................................................... 117
12.2 Riemann Integration on Jordan Regions….............................. 119
12.3 Iterated Integrals… ...................................................................122
12.4 Change of Variables… ............................................................ 125
12.5 Partitions of Unity…................................................................. 130
12.6 The Gamma Function and Volume .......................................... 131
Chapter 13: Fundamental Theorems of Vector Calculus
13.1 Curves… .................................................................................. 135
13.2 Oriented Curves…....................................................................137
13.3 Surfaces…................................................................................ 140
13.4 Oriented Surfaces… ................................................................. 143
13.5 Theorems of Green and Gauss…............................................. 147
13.6 Stokes’s Theorem .................................................................... 150
Chapter 14: Fourier Series
14.1 Introduction… ........................................................................... 156
14.2 Summability of Fourier Series… ............................................... 157
14.3 Growth of Fourier Coefficients… ............................................. 159
14.4 Convergence of Fourier Series… ............................................ 160
14.5 Uniqueness… ........................................................................... 163
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
SOLUTIONS TO EXERCISES
, An Introduction to Analysis
Table of Contents
Chapter 1: The Real Number System
1.2 Ordered field axioms ........................................................... 1
1.3 The Completeness Axiom… ................................................2
1.4 Mathematical Induction… .................................................... 4
1.5 Inverse Functions and Images… ......................................... 6
1.6 Countable and uncountable sets… .....................................8
Chapter 2: Sequences in R
2.1 Limits of Sequences…....................................................... 10
2.2 Limit Theorems...................................................................11
2.3 Bolzano-Weierstrass Theorem ...........................................13
2.4 Cauchy Sequences… ......................................................... 15
2.5 Limits Supremum and Infimum .......................................... 16
Chapter 3: Functions on R
3.1 Two-Sided Limits… ............................................................ 19
3.2 One-Sided Limits and Limits at Infinity… ............................ 20
3.3 Continuity… ........................................................................ 22
3.4 Uniform Continuity… ..........................................................24
Chapter 4: Differentiability on R
4.1 The Derivative… ................................................................. 27
4.2 Differentiability Theorem… ................................................. 28
4.3 The Mean Value Theorem… .............................................. 30
4.4 Taylor’s Theorem and l’Hôpital’s Rule… ........................... 32
4.5 Inverse Function Theorems ................................................ 34
Chapter 5: Integrability on R
5.1 The Riemann Integral… ...................................................... 37
5.2 Riemann Sums .................................................................... 40
5.3 The Fundamental Theorem of Calculus… ...........................43
5.4 Improper Riemann Integration… ........................................ 46
5.5 Functions of Bounded Variation… ...................................... 49
5.6 Convex Functions… ........................................................... 51
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
,Chapter 6: Infinite Series of Real Numbers
6.1 Introduction… .......................................................................53
6.2 Series with Nonnegative Terms… ....................................... 55
6.3 Absolute Convergence…..................................................... 57
6.4 Alternating Series…............................................................. 60
6.5 Estimation of Series… ......................................................... 62
6.6 Additional Tests… ............................................................... 63
Chapter 7: Infinite Series of Functions
7.1 Uniform Convergence of Sequences… ............................... 65
7.2 Uniform Convergence of Series… ....................................... 67
7.3 Power Series… ................................................................... 69
7.4 Analytic Functions… ............................................................72
7.5 Applications…..................................................................... 74
Chapter 8: Euclidean Spaces
8.1 Algebraic Structure… ......................................................... 76
8.2 Planes and Linear Transformations… ................................ 77
8.3 Topology of Rn.......................................................................................................... 79
8.4 Interior, Closure, and Boundary… ...................................... 80
Chapter 9: Convergence in Rn
9.1 Limits of Sequences….........................................................82
9.2 Heine-Borel Theorem .......................................................... 83
9.3 Limits of Functions… ........................................................... 84
9.4 Continuous Functions… ...................................................... 86
9.5 Compact Sets… .................................................................. 87
9.6 Applications…...................................................................... 88
Chapter 10: Metric Spaces
10.1 Introduction… ....................................................................... 90
10.2 Limits of Functions… ............................................................ 91
10.3 Interior, Closure, and Boundary… ........................................ 92
10.4 Compact Sets… ................................................................... 93
10.5 Connected Sets… ................................................................94
10.6 Continuous Functions… ....................................................... 96
10.7 Stone-Weierstrass Theorem................................................. 97
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
, Chapter 11: Differentiability on Rn
11.1 Partial Derivatives and Partial Integrals… .............................. 99
11.2 The Definition of Differentiability…..........................................102
11.3 Derivatives, Differentials, and Tangent Planes… ...................104
11.4 The Chain Rule… ................................................................... 107
11.5 The Mean Value Theorem and Taylor’s Formula…................ 108
11.6 The Inverse Function Theorem .............................................. 111
11.7 Optimization… .........................................................................114
Chapter 12: Integration on Rn
12.1 Jordan Regions….................................................................... 117
12.2 Riemann Integration on Jordan Regions….............................. 119
12.3 Iterated Integrals… ...................................................................122
12.4 Change of Variables… ............................................................ 125
12.5 Partitions of Unity…................................................................. 130
12.6 The Gamma Function and Volume .......................................... 131
Chapter 13: Fundamental Theorems of Vector Calculus
13.1 Curves… .................................................................................. 135
13.2 Oriented Curves…....................................................................137
13.3 Surfaces…................................................................................ 140
13.4 Oriented Surfaces… ................................................................. 143
13.5 Theorems of Green and Gauss…............................................. 147
13.6 Stokes’s Theorem .................................................................... 150
Chapter 14: Fourier Series
14.1 Introduction… ........................................................................... 156
14.2 Summability of Fourier Series… ............................................... 157
14.3 Growth of Fourier Coefficients… ............................................. 159
14.4 Convergence of Fourier Series… ............................................ 160
14.5 Uniqueness… ........................................................................... 163
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
