دانلود کتاب Understanding Analysis

فهرست مطالب :

Preface......Page 6
Contents......Page 12
1.1 Discussion: The Irrationality of 2......Page 14
1.2 Some Preliminaries......Page 18
1.3 The Axiom of Completeness......Page 27
1.4 Consequences of Completeness......Page 33
1.5 Cardinality......Page 38
1.6 Cantor\'s Theorem......Page 45
1.7 Epilogue......Page 49
2.1 Discussion: Rearrangements of Infinite Series......Page 51
2.2 The Limit of a Sequence......Page 54
2.3 The Algebraic and Order Limit Theorems......Page 61
2.4 The Monotone Convergence Theorem and a First Look at Infinite Series......Page 68
2.5 Subsequences and the Bolzano–Weierstrass Theorem......Page 74
2.6 The Cauchy Criterion......Page 78
2.7 Properties of Infinite Series......Page 83
2.8 Double Summations and Products of Infinite Series......Page 91
2.9 Epilogue......Page 95
3.1 Discussion: The Cantor Set......Page 97
3.2 Open and Closed Sets......Page 100
3.3 Compact Sets......Page 108
3.4 Perfect Sets and Connected Sets......Page 114
3.5 Baire\'s Theorem......Page 118
3.6 Epilogue......Page 121
4.1 Discussion: Examples of Dirichlet and Thomae......Page 122
4.2 Functional Limits......Page 126
4.3 Continuous Functions......Page 133
4.4 Continuous Functions on Compact Sets......Page 140
4.5 The Intermediate Value Theorem......Page 147
4.6 Sets of Discontinuity......Page 152
4.7 Epilogue......Page 155
5.1 Discussion: Are Derivatives Continuous?......Page 156
5.2 Derivatives and the Intermediate Value Property......Page 159
5.3 The Mean Value Theorems......Page 166
5.4 A Continuous Nowhere-Differentiable Function......Page 173
5.5 Epilogue......Page 177
6.1 Discussion: The Power of Power Series......Page 179
6.2 Uniform Convergence of a Sequence of Functions......Page 183
6.3 Uniform Convergence and Differentiation......Page 194
6.4 Series of Functions......Page 198
6.5 Power Series......Page 201
6.6 Taylor Series......Page 207
6.7 The Weierstrass Approximation Theorem......Page 215
6.8 Epilogue......Page 221
7.1 Discussion: How Should Integration be Defined?......Page 224
7.2 The Definition of the Riemann Integral......Page 227
7.3 Integrating Functions with Discontinuities......Page 233
7.4 Properties of the Integral......Page 237
7.5 The Fundamental Theorem of Calculus......Page 243
7.6 Lebesgue\'s Criterion for Riemann Integrability......Page 247
7.7 Epilogue......Page 255
8.1 The Generalized Riemann Integral......Page 258
8.2 Metric Spaces and the Baire Category Theorem......Page 267
8.3 Euler\'s Sum......Page 273
8.4 Inventing the Factorial Function......Page 279
8.5 Fourier Series......Page 290
8.6 A Construction of R From Q......Page 306
Bibliography......Page 313
Index......Page 315