Understanding analysis

Cover......Page 1
List of Undergraduate Texts in Mathematics......Page 3
Understanding Analysis......Page 4
Copyright......Page 5
The Main Objectives......Page 6
The Structure of the Book......Page 7
Building a Course......Page 8
The Audience......Page 9
Acknowledgments......Page 10
Contents......Page 12
1.1 Discussion: The Irrationality of √2......Page 14
1.2 Some Preliminaries......Page 17
1.3 The Axiom of Completeness......Page 26
1.4 Consequences of Completeness......Page 31
1.5 Cantor’s Theorem......Page 42
1.6 Epilogue......Page 46
2.1 Discussion: Rearrangements of InfiniteSeries......Page 48
2.2 The Limit of a Sequence......Page 51
2.3 The Algebraic and Order Limit Theorems......Page 57
2.4 The Monotone Convergence Theorem and aFirst Look at Infinite Series......Page 63
2.5 Subsequences and the Bolzano–WeierstrassTheorem......Page 68
2.6 The Cauchy Criterion......Page 71
2.7 Properties of Infinite Series......Page 75
2.8 Double Summations and Productsof Infinite Series......Page 82
2.9 Epilogue......Page 86
3.1 Discussion: The Cantor Set......Page 88
3.2 Open and Closed Sets......Page 91
3.3 Compact Sets......Page 97
3.4 Perfect Sets and Connected Sets......Page 102
3.5 Baire’s Theorem......Page 107
3.6 Epilogue......Page 109
4.1 Discussion: Examples of Dirichletand Thomae......Page 112
4.2 Functional Limits......Page 116
4.3 Combinations of Continuous Functions......Page 122
4.4 Continuous Functions on Compact Sets......Page 127
4.5 The Intermediate Value Theorem......Page 133
4.6 Sets of Discontinuity......Page 138
4.7 Epilogue......Page 140
5.1 Discussion: Are Derivatives Continuous?......Page 142
5.2 Derivatives and the Intermediate Value Property......Page 144
5.3 The Mean Value Theorem......Page 150
5.4 A Continuous Nowhere-Differentiable Function......Page 157
5.5 Epilogue......Page 161
6.1 Discussion: Branching Processes......Page 164
6.2 Uniform Convergence of a Sequence of Functions......Page 167
6.3 Uniform Convergence and Differentiation......Page 177
6.4 Series of Functions......Page 180
6.5 Power Series......Page 182
6.6 Taylor Series......Page 189
6.7 Epilogue......Page 194
7.1 Discussion: How Should Integration be Defined?......Page 196
7.2 The Definition of the Riemann Integral......Page 199
7.3 Integrating Functions with Discontinuities......Page 204
7.4 Properties of the Integral......Page 208
7.5 The Fundamental Theorem of Calculus......Page 212
7.6 Lebesgue’s Criterion for Riemann Integrability......Page 216
7.7 Epilogue......Page 223
8.1 The Generalized Riemann Integral......Page 226
8.2 Metric Spaces and the Baire Category Theorem......Page 235
8.3 Fourier Series......Page 241
8.4 A Construction of RF rom Q......Page 256
Bibliography......Page 264
Index......Page 266
Back Cover......Page 271