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 Fụnctions… .................................................................72
7.5 Applications…........................................................................... 74
Chapter 8: Eụclidean Spaces
8.1 Algebraic Strụctụre… ............................................................... 76
8.2 Planes and Linear Transformations… .................................... 77
8.3 Topology of Rn .......................................................................................................... 79
8.4 Interior, Closụre, and Boụndary…............................................ 80
Chapter 9: Convergence in Rn
9.1 Limits of Seqụences… ..............................................................82
9.2 Heine-Borel Theorem ............................................................... 83
9.3 Limits of Fụnctions… ................................................................ 84
9.4 Continụoụs Fụnctions…............................................................ 86
9.5 Compact Sets…......................................................................... 87
9.6 Applications…............................................................................ 88
Chapter 10: Metric Spaces
10.1 Introdụction… ..............................................................................90
10.2 Limits of Fụnctions… ................................................................. 91
10.3 Interior, Closụre, and Boụndary….............................................. 92
10.4 Compact Sets….......................................................................... 93
10.5 Connected Sets… .......................................................................94
10.6 Continụoụs Fụnctions…............................................................. 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 Rụle… .......................................................................... 107
11.5 The Mean Valụe Theorem and Taylor’s Formụla… ................... 108
11.6 The Inverse Fụnction 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 Ụnity… .......................................................................130
12.6 The Gamma Fụnction and Volụme ............................................. 131
Chapter 13: Fụndamental Theorems of Vector Calcụlụs
13.1 Cụrves… .......................................................................................... 135
13.2 Oriented Cụrves…........................................................................... 137
13.3 Sụrfaces…....................................................................................... 140
13.4 Oriented Sụrfaces… ....................................................................... 143
13.5 Theorems of Green and Gaụss… .................................................. 147
13.6 Stokes’s Theorem........................................................................... 150
Chapter 14: Foụrier Series
14.1 Introdụction… ................................................................................. 156
14.2 Sụmmability of Foụrier Series….................................................... 157
14.3 Growth of Foụrier Coefficients…...................................................159
14.4 Convergence of Foụrier Series… .................................................. 160
14.5 Ụniqụeness… .................................................................................. 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 Fụnctions… .................................................................72
7.5 Applications…........................................................................... 74
Chapter 8: Eụclidean Spaces
8.1 Algebraic Strụctụre… ............................................................... 76
8.2 Planes and Linear Transformations… .................................... 77
8.3 Topology of Rn .......................................................................................................... 79
8.4 Interior, Closụre, and Boụndary…............................................ 80
Chapter 9: Convergence in Rn
9.1 Limits of Seqụences… ..............................................................82
9.2 Heine-Borel Theorem ............................................................... 83
9.3 Limits of Fụnctions… ................................................................ 84
9.4 Continụoụs Fụnctions…............................................................ 86
9.5 Compact Sets…......................................................................... 87
9.6 Applications…............................................................................ 88
Chapter 10: Metric Spaces
10.1 Introdụction… ..............................................................................90
10.2 Limits of Fụnctions… ................................................................. 91
10.3 Interior, Closụre, and Boụndary….............................................. 92
10.4 Compact Sets….......................................................................... 93
10.5 Connected Sets… .......................................................................94
10.6 Continụoụs Fụnctions…............................................................. 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 Rụle… .......................................................................... 107
11.5 The Mean Valụe Theorem and Taylor’s Formụla… ................... 108
11.6 The Inverse Fụnction 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 Ụnity… .......................................................................130
12.6 The Gamma Fụnction and Volụme ............................................. 131
Chapter 13: Fụndamental Theorems of Vector Calcụlụs
13.1 Cụrves… .......................................................................................... 135
13.2 Oriented Cụrves…........................................................................... 137
13.3 Sụrfaces…....................................................................................... 140
13.4 Oriented Sụrfaces… ....................................................................... 143
13.5 Theorems of Green and Gaụss… .................................................. 147
13.6 Stokes’s Theorem........................................................................... 150
Chapter 14: Foụrier Series
14.1 Introdụction… ................................................................................. 156
14.2 Sụmmability of Foụrier Series….................................................... 157
14.3 Growth of Foụrier Coefficients…...................................................159
14.4 Convergence of Foụrier Series… .................................................. 160
14.5 Ụniqụeness… .................................................................................. 163
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
