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.
