Understanding Analysis<br>Stephen Abbott <br><br>Contents<br>Preface v<br>1 The Real Numbers 1<br>1.1 Discussion: The Irrationality of \/2 ................. 1<br>1.2 Some Preliminaries .......................... 4<br>1.3 The Axiom of Completeness ..................... 13<br>1.4 Consequences of Completeness ................... 18<br>1.5 Cantor’s Theorem .......................... 29<br>1.6 Epilogue ................................ 33<br>2 Sequences and Series 35<br>2.1 Discussion: Rearrangements of Infinite Series ........... 35<br>2.2 The Limit of a Sequence ....................... 38<br>2.3 The Algebraic and Order Limit Theorems ............. 44<br>2.4 The Monotone Convergence Theorem and a First Look at<br>Infinite Series ............................. 50<br>2.5 Subsequences and the Bolzano-Weierstrass Theorem ....... 55<br>2.6 The Cauchy Criterion ........................ 58<br>2.7 Properties of Infinite Series ..................... 62<br>2.8 Double Summations and Products of Infinite Series ........ 69<br>2.9 Epilogue ................................ 73<br>3 Basic Topology of R 75<br>3.1 Discussion: The Cantor Set ..................... 75<br>3.2 Open and Closed Sets ........................ 78<br>3.3 Compact Sets ............................. 84<br>3.4 Perfect Sets and Connected Sets .................. 89<br>3.5 Baire’s Theorem ........................... 94<br>3.6 Epilogue ................................ 96<br>4 Functional Limits and Continuity 99<br>4.1 Discussion: Examples of Dirichlet and Thomae .......... 99<br>4.2 Functional Limits ........................... 103<br>4.3 Combinations of Continuous Functions ............... 109<br>XI<br> <br>4.4 Continuous Functions on Compact Sets .............. 114<br>4.5 The Intermediate Value Theorem .................. 120<br>4.6 Sets of Discontinuity ......................... 125<br>4.7 Epilogue ................................ 127<br>5 The Derivative 129<br>5.1 Discussion: Are Derivatives Continuous? .............. 129<br>5.2 Derivatives and the Intermediate Value Property ......... 131<br>5.3 The Mean Value Theorem ...................... 137<br>5.4 A Continuous Nowhere-Differentiable Function .......... 144<br>5.5 Epilogue ................................ 148<br>6 Sequences and Series of Functions 151<br>6.1 Discussion: Branching Processes .................. 151<br>; 6.2 Uniform Convergence of a Sequence of Functions ......... 154<br>6.3 Uniform Convergence and Differentiation ............. 164<br>6.4 Series of Functions .......................... 167<br>6.5 Power Series .............................. 169<br>6.6 Taylor Series ............................. 176<br>6.7 Epilogue ................................ 181<br>7 The Riemann Integral 183<br>7.1 Discussion: How Should Integration be Defined? ......... 183<br>7.2 The Definition of the Riemann Integral............... 186<br>7.3 Integrating Functions with Discontinuities ............. 191<br>7.4 Properties of the Integral ...................... 195<br>7.5 The Fundamental Theorem of Calculus ............... 199<br>7.6 Lebesgue’s Criterion for Riemann Integrability .......... 203<br>7.7 Epilogue ................................ 210<br>8 Additional Topics 213<br>8.1 The Generalized Riemann Integral ................. 213<br>8.2 Metric Spaces and the Baire Category Theorem .......... 222<br>8.3 Fourier Series ............................. 228<br>8.4 A Construction of R From Q .................... 243<br>Bibliography 251<br>Index 253<br>