Covers All 14 Chapters
SOLUTIONS TO EXERCISES
, An Introduction to Analẏsis
Table of Contents
Chapter 1: The Real Number Sẏstem
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 Cauchẏ 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 Infinitẏ… ............................................. 20
3.3 Continuitẏ… .............................................................................................. 22
3.4 Uniform Continuitẏ… ............................................................................... 24
Chapter 4: Differentiabilitẏ on R
4.1 The Derivative…...................................................................................... 27
4.2 Differentiabilitẏ Theorem…....................................................................28
4.3 The Mean Value Theorem… ................................................................. 30
4.4 Taẏlor’s Theorem and l’Hôpital’s Rule… .............................................32
4.5 Inverse Function Theorems ...................................................................... 34
Chapter 5: Integrabilitẏ 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 Analẏtic Functions… ................................................................................. 72
7.5 Applications…........................................................................................... 74
Chapter 8: Euclidean Spaces
8.1 Algebraic Structure… .............................................................................. 76
8.2 Planes and Linear Transformations… .................................................. 77
8.3 Topologẏ of Rn.............................................................................................................................................. 79
8.4 Interior, Closure, and Boundarẏ…........................................................ 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 Boundarẏ…...........................................................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: Differentiabilitẏ on Rn
11.1 Partial Derivatives and Partial Integrals… ............................................. 99
11.2 The Definition of Differentiabilitẏ… .......................................................... 102
11.3 Derivatives, Differentials, and Tangent Planes… .................................. 104
11.4 The Chain Rule… .......................................................................................... 107
11.5 The Mean Value Theorem and Taẏlor’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 Unitẏ…....................................................................................... 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 Summabilitẏ 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 Analẏsis
Table of Contents
Chapter 1: The Real Number Sẏstem
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 Cauchẏ 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 Infinitẏ… ............................................. 20
3.3 Continuitẏ… .............................................................................................. 22
3.4 Uniform Continuitẏ… ............................................................................... 24
Chapter 4: Differentiabilitẏ on R
4.1 The Derivative…...................................................................................... 27
4.2 Differentiabilitẏ Theorem…....................................................................28
4.3 The Mean Value Theorem… ................................................................. 30
4.4 Taẏlor’s Theorem and l’Hôpital’s Rule… .............................................32
4.5 Inverse Function Theorems ...................................................................... 34
Chapter 5: Integrabilitẏ 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 Analẏtic Functions… ................................................................................. 72
7.5 Applications…........................................................................................... 74
Chapter 8: Euclidean Spaces
8.1 Algebraic Structure… .............................................................................. 76
8.2 Planes and Linear Transformations… .................................................. 77
8.3 Topologẏ of Rn.............................................................................................................................................. 79
8.4 Interior, Closure, and Boundarẏ…........................................................ 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 Boundarẏ…...........................................................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: Differentiabilitẏ on Rn
11.1 Partial Derivatives and Partial Integrals… ............................................. 99
11.2 The Definition of Differentiabilitẏ… .......................................................... 102
11.3 Derivatives, Differentials, and Tangent Planes… .................................. 104
11.4 The Chain Rule… .......................................................................................... 107
11.5 The Mean Value Theorem and Taẏlor’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 Unitẏ…....................................................................................... 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 Summabilitẏ 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.
