Understanding Analysis: Second Edition

Main subject categories: • Mathematical Analysis • Real Functions • Real Numbers • Sequences and Series • Basic Topology of R • Functional Limits and Continuity • The Derivative • Sequences and Series of Functions • The Riemann IntegralThis lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one.Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises from the first edition, and three more project-style sections have been added. Investigations of Euler’s computation of ζ(2), the Weierstrass Approximation ­ Theorem, and the gamma function are now among the book's cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis.This is a dangerous book. Understanding Analysis is so well-written and the development of the theory so well-motivated that exposing students to it could well lead them to expect such excellence in all their textbooks... Understanding Analysis is perfectly titled; if your students read it, that's what's going to happen... This terrific book will become the text of choice for the single-variable introductory analysis course... ‒ Steve Kennedy, MAA Reviews 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 This book outlines an elementary, one-semester course that exposes students to both the process of rigor, and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim of a course in real analysis should be to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. This new edition is extensively revised and updated with a refocused layout. In addition to the inclusion of extra exercises, the quality and focus of the exercises in this book has improved, which will help motivate the reader. New features include a discussion of infinite products, and expanded sections on metric spaces, the Baire category theorem, multi-variable functions, and the Gamma function. Reviews from the first edition: "This is a dangerous book. Understanding Analysis is so well-written and the development of the theory so well-motivated that exposing students to it could well lead them to expect such excellence in all their textbooks. ... Understanding Analysis is perfectly titled; if your students read it that s what s going to happen. This terrific book will become the text of choice for the single-variable introductory analysis course; take a look at it next time you re preparing that class." -Steve Kennedy, The Mathematical Association of America, 2001 "Each chapter begins with a discussion section and ends with an epilogue. The discussion serves to motivate the content of the chapter while the epilogue points tantalisingly to more advanced topics. ... I wish I had written this book! The development of the subject follows the tried-and-true path, but the presentation is engaging and challenging. Abbott focuses attention immediately on the topics which make analysis fascinating ... and makes them accessible to an inexperienced audience." -Scott Sciffer, The Australian Mathematical Society Gazette, 29:3, 2002" This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one. Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises from the first edition, and three more project-style sections have been added. Investigations of Euler’s computation of ζ(2), the Weierstrass Approximation Theorem, and the gamma function are now among the book’s cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis. Review of the first edition: “This is a dangerous book. Understanding Analysis is so well-written and the development of the theory so well-motivated t hat exposing students to it could well lead them to expect such excellence in all their textbooks. ... Understanding Analysis is perfectly titled; if your students read it, that’s what’s going to happen. ... This terrific book will become the text of choice for the single-variable introductory analysis course ... ” — Steve Kennedy, MAA Reviews This Lively Introductory Text Exposes The Student To The Rewards Of A Rigorous Study Of Functions Of A Real Variable. In Each Chapter, Informal Discussions Of Questions That Give Analysis Its Inherent Fascination Are Followed By Precise, But Not Overly Formal, Developments Of The Techniques Needed To Make Sense Of Them. By Focusing On The Unifying Themes Of Approximation And The Resolution Of Paradoxes That Arise In The Transition From The Finite To The Infinite, The Text Turns What Could Be A Daunting Cascade Of Definitions And Theorems Into A Coherent And Engaging Progression Of Ideas. Acutely Aware Of The Need For Rigor, The Student Is Much Better Prepared To Understand What Constitutes A Proper Mathematical Proof And How To Write One. The Real Numbers -- Sequences And Series -- Basic Topology Of R -- Functional Limits And Continuity -- The Derivative -- Sequences And Series Of Functions -- The Riemann Integral -- Additional Topics. Stephen Abbott. Includes Bibliographical References (pages 305-306) And Index. Introduction to the Problems in Analysis outlines an elementary, one semester course which exposes students to both the process of rigor, and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim of a course in real analysis should be to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Does the Cantor set contain any irrational numbers? Can the set of points where a function is discontinuous be arbitrary? Can the rational numbers be written as a countable intersection of open sets? Is an infinitely differentiable function necessarily the limit of its Taylor series? Giving these topics center stage, the motivation for a rigorous approach is justified by the fact that they are inaccessible without it.