Covers All 14 Chapters
SOLUTIONS TO EXERCISES
, An Introduction to Analỵsis
Table of Contents
Chapter 1: The Real Number Sỵstem
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 Cauchỵ 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 Infinitỵ… ................................... 20
3.3 Continuitỵ… .................................................................................. 22
3.4 Uniform Continuitỵ… ...................................................................24
Chapter 4: Differentiabilitỵ on R
4.1 The Derivative… ........................................................................... 27
4.2 Differentiabilitỵ Theorem….......................................................... 28
4.3 The Mean Value Theorem…......................................................... 30
4.4 Taỵlor’s Theorem and l’Hôpital’s Rule… ................................... 32
4.5 Inverse Function Theorems ........................................................... 34
Chapter 5: Integrabilitỵ 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
Copỵright © 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 Analỵtic Functions… .................................................................... 72
7.5 Applications…............................................................................... 74
Chapter 8: Euclidean Spaces
8.1 Algebraic Structure….................................................................... 76
8.2 Planes and Linear Transformations… ........................................... 77
8.3 Topologỵ of Rn .................................................................................................................. 79
8.4 Interior, Closure, and Boundarỵ… ................................................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 Boundarỵ… ..................................................92
10.4 Compact Sets…...............................................................................93
10.5 Connected Sets… ........................................................................... 94
10.6 Continuous Functions… .................................................................. 96
10.7 Stone-Weierstrass Theorem ............................................................ 97
Copỵright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
, Chapter 11: Differentiabilitỵ on Rn
11.1 Partial Derivatives and Partial Integrals… ........................................99
11.2 The Definition of Differentiabilitỵ… ............................................... 102
11.3 Derivatives, Differentials, and Tangent Planes… ............................ 104
11.4 The Chain Rule… ............................................................................. 107
11.5 The Mean Value Theorem and Taỵlor’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 Unitỵ… .......................................................................... 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 Summabilitỵ of Fourier Series… ........................................................ 157
14.3 Growth of Fourier Coefficients… ...................................................... 159
14.4 Convergence of Fourier Series… ....................................................... 160
14.5 Uniqueness… ...................................................................................... 163
Copỵright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
SOLUTIONS TO EXERCISES
, An Introduction to Analỵsis
Table of Contents
Chapter 1: The Real Number Sỵstem
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 Cauchỵ 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 Infinitỵ… ................................... 20
3.3 Continuitỵ… .................................................................................. 22
3.4 Uniform Continuitỵ… ...................................................................24
Chapter 4: Differentiabilitỵ on R
4.1 The Derivative… ........................................................................... 27
4.2 Differentiabilitỵ Theorem….......................................................... 28
4.3 The Mean Value Theorem…......................................................... 30
4.4 Taỵlor’s Theorem and l’Hôpital’s Rule… ................................... 32
4.5 Inverse Function Theorems ........................................................... 34
Chapter 5: Integrabilitỵ 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
Copỵright © 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 Analỵtic Functions… .................................................................... 72
7.5 Applications…............................................................................... 74
Chapter 8: Euclidean Spaces
8.1 Algebraic Structure….................................................................... 76
8.2 Planes and Linear Transformations… ........................................... 77
8.3 Topologỵ of Rn .................................................................................................................. 79
8.4 Interior, Closure, and Boundarỵ… ................................................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 Boundarỵ… ..................................................92
10.4 Compact Sets…...............................................................................93
10.5 Connected Sets… ........................................................................... 94
10.6 Continuous Functions… .................................................................. 96
10.7 Stone-Weierstrass Theorem ............................................................ 97
Copỵright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
, Chapter 11: Differentiabilitỵ on Rn
11.1 Partial Derivatives and Partial Integrals… ........................................99
11.2 The Definition of Differentiabilitỵ… ............................................... 102
11.3 Derivatives, Differentials, and Tangent Planes… ............................ 104
11.4 The Chain Rule… ............................................................................. 107
11.5 The Mean Value Theorem and Taỵlor’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 Unitỵ… .......................................................................... 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 Summabilitỵ of Fourier Series… ........................................................ 157
14.3 Growth of Fourier Coefficients… ...................................................... 159
14.4 Convergence of Fourier Series… ....................................................... 160
14.5 Uniqueness… ...................................................................................... 163
Copỵright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
