1 The Real Numbers ............................................. 1
1.1 Discussion: The Irrationality of √2 ..................... 1
1.2 Some Preliminaries ...................................... 5
1.3 The Axiom of Completeness .............................. 14
1.4 Consequences of Completeness ........................... 20
1.5 Cardinality ............................................ 25
1.6 Cantor's Theorem ....................................... 32
1.7 Epilogue ............................................... 36
2 Sequences and Series ........................................ 39
2.1 Discussion: Rearrangements of Infinite Series .......... 39
2.2 The Limit of a Sequence ................................ 42
2.3 The Algebraic and Order Limit Theorems ................. 49
2.4 The Monotone Convergence Theorem and a First Look at
Infinite Series ........................................ 56
2.5 Subsequences and the Bolzano-Weierstrass Theorem ....... 62
2.6 The Cauchy Criterion ................................... 66
2.7 Properties of Infinite Series .......................... 71
2.8 Double Summations and Products of Infinite Series ...... 79
2.9 Epilogue ............................................... 83
3 Basic Topology of R ......................................... 85
3.1 Discussion: The Cantor Set ............................. 85
3.2 Open and Closed Sets ................................... 88
3.3 Compact Sets ........................................... 96
3.4 Perfect Sets and Connected Sets ....................... 102
3.5 Baire's Theorem ....................................... 106
3.6 Epilogue .............................................. 109
4 Functional Limits and Continuity ........................... 111
4.1 Discussion: Examples of Dirichlet and Thomae .......... 111
4.2 Functional Limits ..................................... 115
4.3 Continuous Functions .................................. 122
4.4 Continuous Functions on Compact Sets .................. 129
4.5 The Intermediate Value Theorem ........................ 136
4.6 Sets of Discontinuity ................................. 141
4.7 Epilogue .............................................. 144
5 The Derivative ............................................. 145
5.1 Discussion: Are Derivatives Continuous? ............... 145
5.2 Derivatives and the Intermediate Value Property ....... 148
5.3 The Mean Value Theorems ............................... 155
5.4 A Continuous Nowhere-Differentiable Function .......... 162
5.5 Epilogue .............................................. 166
6 Sequences and Series of Functions .......................... 169
6.1 Discussion: The Power of Power Series ................. 169
6.2 Uniform Convergence of a Sequence of Functions ........ 173
6.3 Uniform Convergence and Differentiation ............... 184
6.4 Series of Functions ................................... 188
6.5 Power Series .......................................... 191
6.6 Taylor Series ......................................... 197
6.7 The Weierstrass Approximation Theorem ................. 205
6.8 Epilogue .............................................. 211
7 The Riemann Integral ...................................... 215
7.1 Discussion: How Should Integration be Defined? ........ 215
7.2 The Definition of the Riemann Integral ................ 218
7.3 Integrating Functions with Discontinuities ............ 224
7.4 Properties of the Integral ............................ 228
7.5 The Fundamental Theorem of Calculus ................... 234
7.6 Lebesgue's Criterion for Riemann Integrability ........ 238
7.7 Epilogue .............................................. 246
8 Additional Topics .......................................... 249
8.1 The Generalized Riemann Integral ...................... 249
8.2 Metric Spaces and the Baire Category Theorem .......... 258
8.3 Euler's Sum ........................................... 264
8.4 Inventing the Factorial Function ...................... 270
8.5 Fourier Series ........................................ 281
8.6 A Construction of R From Q ............................ 297
Bibliography .................................................. 305
Index ......................................................... 307
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