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A Radical Approach to Real Analysis Second Edition AMS / MAA TEXTBOOKS VOL 10 A Radical Approach to Real Analysis Second Edition David Bressoud P1: kpb book3 MAAB001/Bressoud October 20, 2006 4:18 10.1090/text/010 A Radical Approach to Real Analysis © 2007 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2006933946 Hardcover ISBN: 978-0-88385-747-2 (out of print) Paperback ISBN: 978-1-93951-217-8 Electronic ISBN: 978-1-61444-623-1 Printed in the United States of America Current Printing (last digit): 10 9 8 7 6 5 4 3 P1: kpb book3 MAAB001/Bressoud October 20, 2006 4:18 A Radical Approach to Real Analysis Second Edition David M. Bressoud Macalester College ® Published and Distributed by The Mathematical Association of America Committee on Books Frank Farris, Chair MAA Textbooks Editorial Board Zaven A. Karian, Editor George Exner Thomas Garrity Charles R. Hadlock William Higgins Douglas B. Meade Stanley E. Seltzer Shahriar Shahriari Kay B. Somers MAA TEXTBOOKS Bridge to Abstract Mathematics, Ralph W. Oberste-Vorth, Aristides Mouzakitis, and Bonita A. Lawrence Calculus Deconstructed: A Second Course in First-Year Calculus, Zbigniew H. Nitecki Calculus for the Life Sciences: A Modeling Approach, James L. Cornette and Ralph A. Ackerman Combinatorics: A Guided Tour, David R. Mazur Combinatorics: A Problem Oriented Approach, Daniel A. Marcus Common Sense Mathematics, Ethan D. Bolker and Maura B. Mast Complex Numbers and Geometry, Liang-shin Hahn A Course in Mathematical Modeling, Douglas Mooney and Randall Swift Cryptological Mathematics, Robert Edward Lewand Differential Geometry and its Applications, John Oprea Distilling Ideas: An Introduction to Mathematical Thinking, Brian P. Katz and Michael Starbird Elementary Cryptanalysis, Abraham Sinkov Elementary Mathematical Models, Dan Kalman An Episodic History of Mathematics: Mathematical Culture Through Problem Solving, Steven G. Krantz Essentials of Mathematics, Margie Hale Field Theory and its Classical Problems, Charles Hadlock Fourier Series, Rajendra Bhatia Game Theory and Strategy, Philip D. Straffin Geometry Illuminated: An Illustrated Introduction to Euclidean and Hyperbolic Plane Ge- ometry, Matthew Harvey Geometry Revisited, H. S. M. Coxeter and S. L. Greitzer Graph Theory: A Problem Oriented Approach, Daniel Marcus An Invitation to Real Analysis, Luis F. Moreno Knot Theory, Charles Livingston Learning Modern Algebra: From Early Attempts to Prove Fermat’s Last Theorem, Al Cuoco and Joseph J. Rotman The Lebesgue Integral for Undergraduates, William Johnston Lie Groups: A Problem-Oriented Introduction via Matrix Groups, Harriet Pollatsek Mathematical Connections: A Companion for Teachers and Others, Al Cuoco Mathematical Interest Theory, Second Edition, Leslie Jane Federer Vaaler and JamesW. Daniel Mathematical Modeling in the Environment, Charles Hadlock Mathematics for Business Decisions Part 1: Probability and Simulation (electronic text- book), Richard B. Thompson and Christopher G. Lamoureux Mathematics for Business Decisions Part 2: Calculus and Optimization (electronic text- book), Richard B. Thompson and Christopher G. Lamoureux Mathematics for Secondary School Teachers, Elizabeth G. Bremigan, Ralph J. Bremigan, and John D. Lorch The Mathematics of Choice, Ivan Niven The Mathematics of Games and Gambling, Edward Packel Math Through the Ages, William Berlinghoff and Fernando Gouvea Noncommutative Rings, I. N. Herstein Non-Euclidean Geometry, H. S. M. Coxeter Number Theory Through Inquiry, David C. Marshall, Edward Odell, and Michael Starbird Ordinary Differential Equations: from Calculus to Dynamical Systems, V. W. Noonburg A Primer of Real Functions, Ralph P. Boas A Radical Approach to Lebesgue’s Theory of Integration, David M. Bressoud A Radical Approach to Real Analysis, 2nd edition, David M. Bressoud Real Infinite Series, Daniel D. Bonar and Michael Khoury, Jr. Teaching Statistics Using Baseball, 2nd edition, Jim Albert Thinking Geometrically: A Survey of Geometries, Thomas Q. Sibley Topology Now!, Robert Messer and Philip Straffin Understanding our Quantitative World, Janet Andersen and Todd Swanson MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-240-396-5647 P1: kpb book3 MAAB001/Bressoud October 20, 2006 4:18 to the memory of my mother Harriet Carnrite Bressoud P1: kpb book3 MAAB001/Bressoud October 20, 2006 4:18 P1: kpb book3 MAAB001/Bressoud October 20, 2006 4:18 Preface The task of the educator is to make the child’sspirit pass again where its forefathers have gone, mov- ing rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide. —Henri Poincare´ This course of analysis is radical; it returns to the roots of the subject. It is not a history of analysis. It is rather an attempt to follow the injunction of Henri Poincare´ to let history inform pedagogy. It is designed to be a first encounter with real analysis, laying out its context and motivation in terms of the transition from power series to those that are less predictable, especially Fourier series, and marking some of the traps into which even great mathematicians have fallen. This is also an abrupt departure from the standard format and syllabus of analysis. The traditional course begins with a discussion of properties of the real numbers, moves on to continuity, then differentiability, integrability, sequences, and finally infinite series, culminating in a rigorous proof of the properties of Taylor series and perhaps even Fourier series. This is the right way to view analysis, but it is not the right way to teach it. It supplies little motivation for the early definitions and theorems. Careful definitions mean nothing until the drawbacks of the geometric and intuitive understandings of continuity, limits, and series are fully exposed. For this reason, the first part of this book follows the historical progression and moves backwards. It starts with infinite series, illustrating the great successes that led the early pioneers onward, as well as the obstacles that stymied even such luminaries as Euler and Lagrange. There is an intentional emphasis on the mistakes that have been made. These highlight difficult conceptual points. That Cauchy had so much trouble proving the mean value theorem or coming to terms with the notion of uniform convergence should alert us to the fact that these ideas are not easily assimilated. The student needs time with them. The highly refined proofs that we know today leave the mistaken impression that the road of discovery in mathematics is straight and sure. It is not. Experimentation and misunderstanding have been essential components in the growth of mathematics. ix P1: kpb book3 MAAB001/Bressoud October 20, 2006 4:18 x Preface Exploration is an essential component of this course. To facilitate graphical and numerical investigations, Mathematica and Maple commands and programs as well as investigative projects are available on a dedicated website at www.macalester.edu/aratra. The topics considered in this book revolve around the questions raised by Fourier’s trigonometric series and the restructuring of calculus that occurred in the process of an- swering them. Chapter 1 is an introduction to Fourier series: why they are important and why they met with so much resistance. This chapter presupposes familiarity with partial differential equations, but it is purely motivational and can be given as much or as little emphasis as one wishes. Chapter 2 looks at the background to the crisis of 1807. We investigate the difficulties and dangers of working with infinite summations, but also the insights and advances that they make possible. More of these insights and advances are given in Appendix A. Calculus would not have revolutionized mathematics as it did if it had not been coupled with infinite series. Beginning with Newton’s Principia, the physical applications of calculus rely heavily on infinite sums. The chapter concludes with a closer look at the understandings of late eighteenth century mathematicians: how they saw what they were doing and how they justified it. Many of these understandings stood directly in the way of the acceptance of trigonometric series. In Chapter 3, we begin to find answers to the questions raised by Fourier’s series. We follow the efforts of Augustin Louis Cauchy in the 1820s to create a new foundation to the calculus. A careful definition of differentiability comes first, but its application to many of the important questions of the time requires the mean value theorem. Cauchy struggled— unsuccessfully—to prove this theorem. Out of his struggle, an appreciation for the nature of continuity emerges. We return in Chapter 4 to infinite series and investigate the question of convergence. Carl Friedrich Gauss plays an important role through his complete characterization of convergence for the most important class of power series: the hypergeometric series. This chapter concludes with a verification that the Fourier cosine series studied in the first chapter does, in fact, converge at every value of x. The strange behavior of infinite sums of functions is finally tackled in Chapter 5. We look at Dirichlet’s insights into the problems associated with grouping and rearranging infinite series. We watch Cauchy as he wrestles with the problem of the discontinuity of an infinite sum of continuous functions, and we discover the key that he was missing. We begin to answer the question of when it is legitimate to differentiate or integrate an infinite series by differentiating or integrating
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