Solutions Manual
Understanding Analysis
By Abbott
2nd Edition
,Contents
Preface V
1 The Real Numbers √ 1
1.1 Discussion: The Irrationality 2 .......................................................... 1
Of
1.2 Some Preliminaries ..................................................................................... 4
1.3 The Axiom Of Completeness ................................................................... 13
1.4 Consequences Of Completeness ............................................................... 18
1.5 Cantor’s Theorem ..................................................................................... 29
1.6 Epilogue ...................................................................................................... 33
2 Sequences And Series 35
2.1 Discussion: Rearrangements Of Infinite Series ...................................... 35
2.2 The Limit Of A Sequence ......................................................................... 38
2.3 The Algebraic And Order Limit Theorems ........................................... 44
2.4 The Monotone Convergence Theorem And A First Look At
Infinite Series............................................................................................. 50
2.5 Subsequences And The Bolzano–Weierstrass Theorem ........................ 55
2.6 The Cauchy Criterion ............................................................................... 58
2.7 Properties Of Infinite Series..................................................................... 62
2.8 Double Summations And Products Of Infinite Series ........................... 69
2.9 Epilogue ...................................................................................................... 73
3 Basic Topology Of R 75
3.1 Discussion: The Cantor Set ..................................................................... 75
3.2 Open And Closed Sets .............................................................................. 78
3.3 Compact Sets ........................................................................................... 84
3.4 Perfect Sets And Connected Sets ............................................................ 89
3.5 Baire’s Theorem ........................................................................................ 94
3.6 Epilogue ...................................................................................................... 96
4 Functional Limits And Continuity 99
4.1 Discussion: Examples Of Dirichlet And Thomae ................................... 99
4.2 Functional Limits .................................................................................... 103
4.3 Combinations Of Continuous Functions ............................................... 109
Xi
, 4.4 Continuous Functions On Compact Sets .............................................. 114
4.5 The Intermediate Value Theorem ......................................................... 120
4.6 Sets Of Discontinuity ............................................................................... 125
4.7 Epilogue .................................................................................................... 127
5 The Derivative 129
5.1 Discussion: Are Derivatives Continuous? ............................................. 129
5.2 Derivatives And The Intermediate Value Property ............................. 131
5.3 The Mean Value Theorem ..................................................................... 137
5.4 A Continuous Nowhere-Differentiable Function .................................. 144
5.5 Epilogue .................................................................................................... 148
6 Sequences And Series Of Functions 151
6.1 Discussion: Branching Processes .......................................................... 151
6.2 Uniform Convergence Of A Sequence Of Functions ............................. 154
6.3 Uniform Convergence And Differentiation ........................................... 164
6.4 Series Of Functions .................................................................................. 167
6.5 Power Series............................................................................................. 169
6.6 Taylor Series ............................................................................................ 176
6.7 Epilogue .................................................................................................... 181
7 The Riemann Integral 183
7.1 Discussion: How Should Integration Be Defined? ............................... 183
7.2 The Definition Of The Riemann Integral ............................................. 186
7.3 Integrating Functions With Discontinuities ......................................... 191
7.4 Properties Of The Integral...................................................................... 195
7.5 The Fundamental Theorem Of Calculus ............................................... 199
7.6 Lebesgue’s Criterion For Riemann Integrability .................................. 203
7.7 Epilogue .................................................................................................... 210
8 Additional Topics 213
8.1 The Generalized Riemann Integral ....................................................... 213
8.2 Metric Spaces And The Baire Category Theorem............................... 222
8.3 Fourier Series ........................................................................................... 228
8.4 A Construction Of R From Q............................................................... 243
Bibliography 251
Index 253
, xii Contents
Chapter 1
The Real Numbers
√
1.1 Discussion: The Irrationality Of 2
Toward The End Of His Distinguished Career, The Renowned British
Mathematician
G.H. Hardy Eloquently Laid Out A Justification For A Life Of Studying
Mathematics In A Mathematician’s Apology, An Essay First Published In
1940. At The Center Of Hardy’s Defense Is The Thesis That Mathematics Is An
Aesthetic Discipline. For Hardy, The Applied Mathematics Of Engineers And
Economists Held Little Charm. “Real Mathematics,” As He Referred To It,
“Must Be Justified As Art If It Can Be Justified At All.”
To Help Make His Point, Hardy Includes Two Theorems From Classical Greek
Mathematics, Which, In His Opinion, Possess An Elusive Kind Of Beauty That,
Although Difficult To Define, Is Easy To Recognize. The First Of These Results Is
Euclid’s Proof That There Are An Infinite Number Of Prime Numbers. The
Second
Result Is T h √E Discovery, Attributed To The School Of Pythagoras From Around
500
B.C., That 2 Is Irrational. It Is This Second Theorem That Demands Our
Attention.
(A Course In Number Theory Would Focus On The First.) The Argument Uses
Only Arithmetic, But Its Depth And Importance Cannot Be Overstated. As Hardy
Says, “[It] Is A ‘Simple’ Theorem, Simple Both In Idea And Execution, But There
Is No Doubt At All About [It Being] Of The Highest Class. [It] Is As Fresh And
Significant As When It Was Discovered—Two Thousand Years Have Not Written
A Wrinkle On [It].”
Theorem 1.1.1. There Is No Rational Number Whose Square Is 2.
Proof. A Rational Number Is Any Number That Can Be Expressed In The Form
P/Q, Where P And Q Are Integers. Thus, What The Theorem Asserts Is That No
Matter How P And Q Are Chosen, It Is Never The Case That (P/Q)2 = 2. The
Line Of Attack Is Indirect, Using A Type Of Argument Referred To As A
Proof By Contradiction. The Idea Is To Assume That There Is A Rational
Number Whose Square Is 2 And Then Proceed Along Logical Lines Until We
Understanding Analysis
By Abbott
2nd Edition
,Contents
Preface V
1 The Real Numbers √ 1
1.1 Discussion: The Irrationality 2 .......................................................... 1
Of
1.2 Some Preliminaries ..................................................................................... 4
1.3 The Axiom Of Completeness ................................................................... 13
1.4 Consequences Of Completeness ............................................................... 18
1.5 Cantor’s Theorem ..................................................................................... 29
1.6 Epilogue ...................................................................................................... 33
2 Sequences And Series 35
2.1 Discussion: Rearrangements Of Infinite Series ...................................... 35
2.2 The Limit Of A Sequence ......................................................................... 38
2.3 The Algebraic And Order Limit Theorems ........................................... 44
2.4 The Monotone Convergence Theorem And A First Look At
Infinite Series............................................................................................. 50
2.5 Subsequences And The Bolzano–Weierstrass Theorem ........................ 55
2.6 The Cauchy Criterion ............................................................................... 58
2.7 Properties Of Infinite Series..................................................................... 62
2.8 Double Summations And Products Of Infinite Series ........................... 69
2.9 Epilogue ...................................................................................................... 73
3 Basic Topology Of R 75
3.1 Discussion: The Cantor Set ..................................................................... 75
3.2 Open And Closed Sets .............................................................................. 78
3.3 Compact Sets ........................................................................................... 84
3.4 Perfect Sets And Connected Sets ............................................................ 89
3.5 Baire’s Theorem ........................................................................................ 94
3.6 Epilogue ...................................................................................................... 96
4 Functional Limits And Continuity 99
4.1 Discussion: Examples Of Dirichlet And Thomae ................................... 99
4.2 Functional Limits .................................................................................... 103
4.3 Combinations Of Continuous Functions ............................................... 109
Xi
, 4.4 Continuous Functions On Compact Sets .............................................. 114
4.5 The Intermediate Value Theorem ......................................................... 120
4.6 Sets Of Discontinuity ............................................................................... 125
4.7 Epilogue .................................................................................................... 127
5 The Derivative 129
5.1 Discussion: Are Derivatives Continuous? ............................................. 129
5.2 Derivatives And The Intermediate Value Property ............................. 131
5.3 The Mean Value Theorem ..................................................................... 137
5.4 A Continuous Nowhere-Differentiable Function .................................. 144
5.5 Epilogue .................................................................................................... 148
6 Sequences And Series Of Functions 151
6.1 Discussion: Branching Processes .......................................................... 151
6.2 Uniform Convergence Of A Sequence Of Functions ............................. 154
6.3 Uniform Convergence And Differentiation ........................................... 164
6.4 Series Of Functions .................................................................................. 167
6.5 Power Series............................................................................................. 169
6.6 Taylor Series ............................................................................................ 176
6.7 Epilogue .................................................................................................... 181
7 The Riemann Integral 183
7.1 Discussion: How Should Integration Be Defined? ............................... 183
7.2 The Definition Of The Riemann Integral ............................................. 186
7.3 Integrating Functions With Discontinuities ......................................... 191
7.4 Properties Of The Integral...................................................................... 195
7.5 The Fundamental Theorem Of Calculus ............................................... 199
7.6 Lebesgue’s Criterion For Riemann Integrability .................................. 203
7.7 Epilogue .................................................................................................... 210
8 Additional Topics 213
8.1 The Generalized Riemann Integral ....................................................... 213
8.2 Metric Spaces And The Baire Category Theorem............................... 222
8.3 Fourier Series ........................................................................................... 228
8.4 A Construction Of R From Q............................................................... 243
Bibliography 251
Index 253
, xii Contents
Chapter 1
The Real Numbers
√
1.1 Discussion: The Irrationality Of 2
Toward The End Of His Distinguished Career, The Renowned British
Mathematician
G.H. Hardy Eloquently Laid Out A Justification For A Life Of Studying
Mathematics In A Mathematician’s Apology, An Essay First Published In
1940. At The Center Of Hardy’s Defense Is The Thesis That Mathematics Is An
Aesthetic Discipline. For Hardy, The Applied Mathematics Of Engineers And
Economists Held Little Charm. “Real Mathematics,” As He Referred To It,
“Must Be Justified As Art If It Can Be Justified At All.”
To Help Make His Point, Hardy Includes Two Theorems From Classical Greek
Mathematics, Which, In His Opinion, Possess An Elusive Kind Of Beauty That,
Although Difficult To Define, Is Easy To Recognize. The First Of These Results Is
Euclid’s Proof That There Are An Infinite Number Of Prime Numbers. The
Second
Result Is T h √E Discovery, Attributed To The School Of Pythagoras From Around
500
B.C., That 2 Is Irrational. It Is This Second Theorem That Demands Our
Attention.
(A Course In Number Theory Would Focus On The First.) The Argument Uses
Only Arithmetic, But Its Depth And Importance Cannot Be Overstated. As Hardy
Says, “[It] Is A ‘Simple’ Theorem, Simple Both In Idea And Execution, But There
Is No Doubt At All About [It Being] Of The Highest Class. [It] Is As Fresh And
Significant As When It Was Discovered—Two Thousand Years Have Not Written
A Wrinkle On [It].”
Theorem 1.1.1. There Is No Rational Number Whose Square Is 2.
Proof. A Rational Number Is Any Number That Can Be Expressed In The Form
P/Q, Where P And Q Are Integers. Thus, What The Theorem Asserts Is That No
Matter How P And Q Are Chosen, It Is Never The Case That (P/Q)2 = 2. The
Line Of Attack Is Indirect, Using A Type Of Argument Referred To As A
Proof By Contradiction. The Idea Is To Assume That There Is A Rational
Number Whose Square Is 2 And Then Proceed Along Logical Lines Until We
