Problems in Real Analysis

Transcription

1 Problems in Real Analysis

2 Teodora-Liliana T. Rădulescu Vicenţiu D. Rădulescu Titu Andreescu Problems in Real Analysis Advanced Calculus on the Real Axis

3 Teodora-Liliana T. Rădulescu Department of Mathematics Fratii Buzesti National College Craiova Romania Vicenţiu D. Rădulescu Simion Stoilow Mathematics Institute Romanian Academy Bucharest Romania Titu Andreescu School of Natural Sciences and Mathematics University of Texas at Dallas Richardson, TX USA ISBN: e-isbn: DOI: / Springer Dordrecht Heidelberg London New York Library of Congress Control Number: Mathematics Subject Classification (2000): 00A07, 26-01, 28-01, Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (

4 To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems. George Pólya ( ) We come nearest to the great when we are great in humility. Rabindranath Tagore ( )

5 Foreword This carefully written book presents an extremely motivating and original approach, by means of problem-solving, to calculus on the real line, and as such, serves as a perfect introduction to real analysis. To achieve their goal, the authors have carefully selected problems that cover an impressive range of topics, all at the core of the subject. Some problems are genuinely difficult, but solving them will be highly rewarding, since each problem opens a new vista in the understanding of mathematics. This book is also perfect for self-study, since solutions are provided. I like the care with which the authors intersperse their text with careful reviews of the background material needed in each chapter, thought-provoking quotations, and highly interesting and well-documented historical notes. In short, this book also makes very pleasant reading, and I am confident that each of its readers will enjoy reading it as much as I did. The charm and never-ending beauty of mathematics pervade all its pages. In addition, this little gem illustrates the idea that one cannot learn mathematics without solving difficult problems. It is a world apart from the computer addiction that we are unfortunately witnessing among the younger generations of would-be mathematicians, who use too much ready-made software instead or their brains, or who stand in awe in front of computer-generated images, as if they had become the essence of mathematics. As such, it carries a very useful message. One cannot help comparing this book to a great ancestor, the famed Problems and Theorems in Analysis, bypólya and Szegő, a text that has strongly influenced generations of analysts. I am confident that this book will have a similar impact. Hong Kong, July 2008 Philippe G. Ciarlet vii

6 Preface If I have seen further it is by standing on the shoulders of giants. Sir Isaac Newton ( ), Letter to Robert Hooke, 1675 Mathematical analysis is central to mathematics, whether pure or applied. This discipline arises in various mathematical models whose dependent variables vary continuously and are functions of one or several variables. Real analysis dates to the mid-nineteenth century, and its roots go back to the pioneering papers by Cauchy, Riemann, and Weierstrass. In 1821, Cauchy established new requirements of rigor in his celebrated Cours d Analyse. The questions he raised are the following: What is a derivative really? Answer: a limit. What is an integral really? Answer: a limit. What is an infinite series really? Answer: a limit. This leads to What is a limit? Answer: a number. And, finally, the last question: What is a number? Weierstrass and his collaborators (Heine, Cantor) answered this question around Our treatment in this volume is strongly related to the pioneering contributions in differential calculus by Newton, Leibniz, Descartes, and Euler in the seventeenth and eighteenth centuries, with mathematical rigor in the nineteenth century promoted by Cauchy, Weierstrass, and Peano. This presentation furthers modern directions in the integral calculus developed by Riemann and Darboux. Due to the huge impact of mathematical analysis, we have intended in this book to build a bridge between ordinary high-school or undergraduate exercises and more difficult and abstract concepts or problems related to this field. We present in this volume an unusual collection of creative problems in elementary mathematical analysis. We intend to develop some basic principles and solution techniques and to offer a systematic illustration of how to organize the natural transition from problemsolving activity toward exploring, investigating, and discovering new results and properties. ix

7 x Preface The aim of this volume in elementary mathematical analysis is to introduce, through problems-solving, fundamental ideas and methods without losing sight of the context in which they first developed and the role they play in science and particularly in physics and other applied sciences. This volume aims at rapidly developing differential and integral calculus for real-valued functions of one real variable, giving relevance to the discussion of some differential equations and maximum principles. The book is mainly geared toward students studying the basic principles of mathematical analysis. However, given its selection of problems, organization, and level, it would be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam and other high-level mathematical contests. We also address this work to motivated high-school and undergraduate students. This volume is meant primarily for students in mathematics, physics, engineering, and computer science, but, not without authorial ambition, we believe it can be used by anyone who wants to learn elementary mathematical analysis by solving problems. The book is also a must-have for instructors wishing to enrich their teaching with some carefully chosen problems and for individuals who are interested in solving difficult problems in mathematical analysis on the real axis. The volume is intended as a challenge to involve students as active participants in the course. To make our work self-contained, all chapters include basic definitions and properties. The problems are clustered by topic into eight chapters, each of them containing both sections of proposed problems with complete solutions and separate sections including auxiliary problems, their solutions being left to our readers. Throughout the book, students are encouraged to express their own ideas, solutions, generalizations, conjectures, and conclusions. The volume contains a comprehensive collection of challenging problems, our goal being twofold: first, to encourage the readers to move away from routine exercises and memorized algorithms toward creative solutions and nonstandard problem-solving techniques; and second, to help our readers to develop a host of new mathematical tools and strategies that will be useful beyond the classroom and in a number of applied disciplines. We include representative problems proposed at various national or international competitions, problems selected from prestigious mathematical journals, but also some original problems published in leading publications. That is why most of the problems contained in this book are neither standard nor easy. The readers will find both classical topics of mathematical analysis on the real axis and modern ones. Additionally, historical comments and developments are presented throughout the book in order to stimulate further inquiry. Traditionally, a rigorous first course or problem book in elementary mathematical analysis progresses in the following order: Sequences Functions = Continuity = Differentiability= Integration Limits

8 Preface xi However, the historical development of these subjects occurred in reverse order: Cauchy (1821) = Weierstrass (1872) = Archimedes Newton (1665) Leibniz (1675) = Kepler (1615) Fermat (1638) This book brings to life the connections among different areas of mathematical analysis and explains how various subject areas flow from one another. The volume illustrates the richness of elementary mathematical analysis as one of the most classical fields in mathematics. The topic is revisited from the higher viewpoint of university mathematics, presenting a deeper understanding of familiar subjects and an introduction to new and exciting research fields, such as Ginzburg Landau equations, the maximum principle, singular differential and integral inequalities, and nonlinear differential equations. The volume is divided into four parts, ten chapters, and two appendices, as follows: Part I. Sequences, Series, and Limits Chapter 1. Sequences Chapter 2. Series Chapter 3. Limits of Functions Part II. Qualitative Properties of Continuous and Differentiable Functions Chapter 4. Continuity Chapter 5. Differentiability Part III. Applications to Convex Functions and Optimization Chapter 6. Convex Functions Chapter 7. Inequalities and Extremum Problems Part IV. Antiderivatives, Riemann Integrability, and Applications Chapter 8. Antiderivatives Chapter 9. Riemann Integrability Chapter 10. Applications of the Integral Calculus Appendix A. Basic Elements of Set Theory Appendix B. Topology of the Real Line Each chapter is divided into sections. Exercises, formulas, and figures are numbered consecutively in each section, and we also indicate both the chapter and the section numbers. We have included at the beginning of chapters and sections quotations from the literature. They are intended to give the flavor of mathematics as a science with a long history. This book also contains a rich glossary and index, as well as a list of abbreviations and notation.

9 xii Preface Key features of this volume: contains a collection of challenging problems in elementary mathematical analysis; includes incisive explanations of every important idea and develops illuminating applications of many theorems, along with detailed solutions, suitable crossreferences, specific how-to hints, and suggestions; is self-contained and assumes only a basic knowledge but opens the path to competitive research in the field; uses competition-like problems as a platform for training typical inventive skills; develops basic valuable techniques for solving problems in mathematical analysis on the real axis; 38 carefully drawn figures support the understanding of analytic concepts; includes interesting and valuable historical account of ideas and methods in analysis; contains excellent bibliography, glossary, and index. The book has elementary prerequisites, and it is designed to be used for lecture courses on methodology of mathematical research or discovery in mathematics. This work is a first step toward developing connections between analysis and other mathematical disciplines, as well as physics and engineering. The background the student needs to read this book is quite modest. Anyone with elementary knowledge in calculus is well-prepared for almost everything to be found here. Taking into account the rich introductory blurbs provided with each chapter, no particular prerequisites are necessary, even if a dose of mathematical sophistication is needed. The book develops many results that are rarely seen, and even experienced readers are likely to find material that is challenging and informative. Our vision throughout this volume is closely inspired by the following words of George Pólya [90] (1945) on the role of problems and discovery in mathematics: Infallible rules of discovery leading to the solution of all possible mathematical problems would be more desirable than the philosopher s stone, vainly sought by all alchemists. The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea. Those of us who have little luck and less brain sometimes sit for decades. The fact seems to be, as Poincaré observed, it is the man, not the method, that solves the problem. Despite our best intentions, errors are sure to have slipped by us. Please let us know of any you find. August 2008 Teodora-Liliana Rădulescu Vicenţiu Rădulescu Titu Andreescu

10 Acknowledgments We acknowledge, with unreserved gratitude, the crucial role of Professors Catherine Bandle, Wladimir-Georges Boskoff, Louis Funar, Patrizia Pucci, Richard Stong, and Michel Willem, who encouraged us to write a problem book on this subject. Our colleague and friend Professor Dorin Andrica has been very interested in this project and suggested some appropriate problems for this volume. We warmly thank Professors Ioan Şerdean and Marian Tetiva for their kind support and useful discussions. This volume was completed while Vicenţiu Rădulescu was visiting the University of Ljubljana during July and September 2008 with a research position funded by theslovenianresearch Agency. Hewouldlike to thankprofessordušan Repovš for the invitation and many constructive discussions. We thank Dr. Nicolae Constantinescu and Dr. Mirel Coşulschi for the professional drawing of figures contained in this book. We are greatly indebted to the anonymous referees for their careful reading of the manuscript and for numerous comments and suggestions. These precious constructive remarks were very useful to us in the elaboration of the final version of this volume. We are grateful to Ann Kostant, Springer editorial director for mathematics, for her efficient and enthusiastic help, as well as for numerous suggestions related to previous versions of this book. Our special thanks go also to Laura Held and to the other members of the editorial technical staff of Springer New York for the excellent quality of their work. We are particularly grateful to copyeditor David Kramer for his guidance, thoroughness and attention to detail. V. Rădulescu acknowledges the support received from the Romanian Research Council CNCSIS under Grant 55/2008 Sisteme diferenţiale în analiza neliniară şi aplicaţii. xiii

11 Contents Foreword... Preface... vii ix Acknowledgments... xiii Abbreviations and Notation... xix Part I Sequences, Series, and Limits 1 Sequences MainDefinitionsandBasicResults Introductory Problems RecurrentSequences Qualitative Results Hardy s and Carleman s Inequalities IndependentStudyProblems Series MainDefinitionsandBasicResults ElementaryProblems ConvergentandDivergentSeries Infinite Products Qualitative Results IndependentStudyProblems Limits of Functions MainDefinitionsandBasicResults ComputingLimits Qualitative Results IndependentStudyProblems xv

12 xvi Contents Part II Qualitative Properties of Continuous and Differentiable Functions 4 Continuity TheConceptofContinuityandBasicProperties ElementaryProblems TheIntermediateValueProperty Types of Discontinuities FixedPoints Functional Equations and Inequalities Qualitative Properties of Continuous Functions IndependentStudyProblems Differentiability TheConceptofDerivativeandBasicProperties Introductory Problems TheMainTheorems TheMaximumPrinciple Differential Equations and Inequalities IndependentStudyProblems Part III Applications to Convex Functions and Optimization 6 Convex Functions MainDefinitionsandBasicResults BasicPropertiesofConvexFunctionsandApplications Convexity versus Continuity and Differentiability Qualitative Results IndependentStudyProblems Inequalities and Extremum Problems BasicTools ElementaryExamples Jensen, Young, Hölder, Minkowski, and Beyond OptimizationProblems Qualitative Results IndependentStudyProblems Part IV Antiderivatives, Riemann Integrability, and Applications 8 Antiderivatives MainDefinitionsandProperties ElementaryExamples ExistenceorNonexistenceofAntiderivatives Qualitative Results IndependentStudyProblems...324

13 Contents xvii 9 Riemann Integrability MainDefinitionsandProperties ElementaryExamples ClassesofRiemannIntegrableFunctions BasicRulesforComputingIntegrals RiemannIintegralsandLimits Qualitative Results IndependentStudyProblems Applications of the Integral Calculus Overview Integral Inequalities ImproperIntegrals IntegralsandSeries ApplicationstoGeometry IndependentStudyProblems Part V Appendix A Basic Elements of Set Theory A.1 DirectandInverseImageofaSet A.2 Finite, Countable, and Uncountable Sets B Topology of the Real Line B.1 OpenandClosedSets B.2 Some Distinguished Points Glossary References Index...443

14 Abbreviations and Notation Abbreviations We have tried to avoid using nonstandard abbreviations as much as possible. Other abbreviations include: AMM American Mathematical Monthly GMA Mathematics Gazette, Series A MM Mathematics Magazine IMO International Mathematical Olympiad IMCUS International Mathematics Competition for University Students MSC Miklós Schweitzer Competitions Putnam The William Lowell Putnam Mathematical Competition SEEMOUS South Eastern European Mathematical Olympiad for University Students Notation We assume familiarity with standard elementary notation of set theory, logic, algebra, analysis, number theory, and combinatorics. The following is notation that deserves additional clarification. N the set of nonnegative integers (N = {0,1,2,3,...}) N the set of positive integers (N = {1,2,3,...}) Z the set of integer real numbers (Z = {..., 3, 2, 1,0,1,2,3,...}) Z the set of nonzero integer real numbers (Z = Z \{0}) Q the ( set{ of rational real numbers Q = mn ; m Z, n N, m and n are relatively prime }) R the set of real numbers R the set of nonzero real numbers (R = R \{0}) R + R the set of nonnegative real numbers (R + =[0,+ )) the completed real line ( R = R {,+ } ) xix

15 xx Abbreviations and Notation C the set of( complex numbers e lim n n n) = supa the least upper bound of the set A R infa the greatest lower bound of the set A R x + the positive part of the real number x (x + = max{x,0}) x the negative part of the real number x (x = max{ x,0}) x the modulus (absolute value) of the real number x ( x = x + + x ) {x} the fractional part of the real number x (x =[x]+{x}) Card(A) cardinality of the finite set A dist(x,a) the distance from x R to the set A R (dist(x,a)=inf{ x a ; a A}) IntA the set of interior points of A R f (A) the image of the set A under a mapping f f 1 (B) the inverse image of the set B under a mapping f f g the composition of functions f and g: ( f g)(x)= f (g(x)) n! n factorial, equal to n(n 1) 1 (n N ) (2n)!! 2n(2n 2)(2n 4) 4 2 (n N ) (2n + 1)!! (2n + 1)(2n 1)(2n 3) 3 1 (n N ) lnx log e x (x > 0) x x 0 x x 0 R and x < x 0 x x 0 x x 0 ( R and ) x > x 0 lim supx n lim supx k n n k n ( ) lim inf n n lim n inf k k n f (n) (x) nth derivative of the function f at x C n (a,b) the set of n-times differentiable functions f : (a,b) R such that f (n) is continuous on (a,b) C (a,b) the set of infinitely differentiable functions f : (a,b) R (C (a,b)= n=0 C n (a,b)) Δ f the ( Laplace operator applied to the function f : D R N R Δ f = 2 f + 2 f ) f x 2 1 x 2 2 x 2 N Landau s notation f (x)=o(g(x)) as x x 0 if f (x)/g(x) 0asx x 0 f (x)=o(g(x)) as x x 0 if f (x)/g(x) is bounded in a neighborhood of x 0 f g as x x 0 if f (x)/g(x) 1 asx x 0 Hardy s notation f g as x x 0 if f (x)/g(x) 0asx x 0 f g as x x 0 if f (x)/g(x) is bounded in a neighborhood of x 0