Statistics R A BASIC COURSE IN AA BBAASSIICC CCOOUURRSSEE IINN REAL ANALYSIS EA RREEAALL A B LA S AANNAALLYYSSIISS Based on the authors’ combined 35 years of experience in teaching, I AC A Basic Course in Real Analysis introduces students to the aspects of real analysis in a friendly way. The authors offer insights into the NC O way a typical mathematician works observing patterns, conducting AU experiments by means of looking at or creating examples, trying to R L understand the underlying principles, and coming up with guesses S or conjectures and then proving them rigorously based on his or her YE explorations. SI N With more than 100 pictures, the book creates interest in real analysis I S by encouraging students to think geometrically. Each difficult proof is prefaced by a strategy and explanation of how the strategy is translated into rigorous and precise proofs. The authors then explain the mystery and role of inequalities in analysis to train students to arrive at estimates that will be useful for proofs. They highlight the role K of the least upper bound property of real numbers, which underlies U M all crucial results in real analysis. In addition, the book demonstrates A analysis as a qualitative as well as quantitative study of functions, R exposing students to arguments that fall under hard analysis. • K Although there are many books available on this subject, students U often find it difficult to learn the essence of analysis on their own or after M A going through a course on real analysis. Written in a conversational R AAJJIITT KKUUMMAARR tone, this book explains the hows and whys of real analysis and E provides guidance that makes readers think at every stage. S A SS.. KKUUMMAARREESSAANN N K22053 K22053_Cover.indd 1 12/6/13 9:08 AM A BASIC COURSE IN REAL ANALYSIS A BASIC COURSE IN REAL ANALYSIS AJIT KUMAR S. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Dedicated to all who believe and take pride in the dignity of teaching and especially to the Mathematics Training and Talent Search Programme, India Contents Preface xi To the Students xiii About the Authors xv List of Figures xvii 1 Real Number System 1 1.1 Algebra of the Real Number System . . . . . . . . . . . . . . . . . 1 1.2 Upper and Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 LUB Property and Its Applications . . . . . . . . . . . . . . . . . . 7 1.4 Absolute Value and Triangle Inequality . . . . . . . . . . . . . . . 20 2 Sequences and Their Convergence 27 2.1 Sequences and Their Convergence. . . . . . . . . . . . . . . . . . . 28 2.2 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3 Monotone Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4 Sandwich Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5 Some Important Limits . . . . . . . . . . . . . . . . . . . . . . . . 48 2.6 Sequences Diverging to ±∞ . . . . . . . . . . . . . . . . . . . . . . 52 2.7 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.8 Sequences Defined Recursively . . . . . . . . . . . . . . . . . . . . 58 3 Continuity 63 3.1 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2 ε-δ Definition of Continuity . . . . . . . . . . . . . . . . . . . . . . 71 3.3 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . . 78 3.4 Extreme Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . 84 3.5 Monotone Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.6 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.7 Uniform Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.8 Continuous Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 103 vii viii CONTENTS 4 Differentiation 109 4.1 Differentiability of Functions . . . . . . . . . . . . . . . . . . . . . 110 4.2 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.3 L’Hospital’s Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.4 Higher-order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 134 4.5 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.6 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.7 Cauchy’s Form of the Remainder . . . . . . . . . . . . . . . . . . . 150 5 Infinite Series 153 5.1 Convergence of an Infinite Series . . . . . . . . . . . . . . . . . . . 154 5.2 Abel’s Summation by Parts . . . . . . . . . . . . . . . . . . . . . . 163 5.3 Rearrangements of an Infinite Series . . . . . . . . . . . . . . . . . 165 5.4 Cauchy Product of Two Infinite Series . . . . . . . . . . . . . . . . 172 6 Riemann Integration 175 6.1 Darboux Integrability . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.2 Properties of the Integral . . . . . . . . . . . . . . . . . . . . . . . 186 6.3 Fundamental Theorems of Calculus . . . . . . . . . . . . . . . . . . 194 6.4 Mean Value Theorems for Integrals . . . . . . . . . . . . . . . . . . 199 6.5 Integral Form of the Remainder . . . . . . . . . . . . . . . . . . . . 203 6.6 Riemann’s Original Definition . . . . . . . . . . . . . . . . . . . . . 205 6.7 Sum of an Infinite Series as an Integral . . . . . . . . . . . . . . . . 210 6.8 Logarithmic and Exponential Functions . . . . . . . . . . . . . . . 212 6.9 Improper Riemann Integrals . . . . . . . . . . . . . . . . . . . . . . 214 7 Sequences and Series of Functions 221 7.1 Pointwise Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 221 7.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 228 7.3 Consequences of Uniform Convergence . . . . . . . . . . . . . . . . 231 7.4 Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 7.5 Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 7.6 Taylor Series of a Smooth Function . . . . . . . . . . . . . . . . . . 258 7.7 Binomial Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 7.8 Weierstrass Approximation Theorem . . . . . . . . . . . . . . . . . 264 A Quantifiers 271 B Limit Inferior and Limit Superior 277 C Topics for Student Seminars 283 CONTENTS ix D Hints for Selected Exercises 287 D.1 Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 D.2 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 D.3 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 D.4 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 D.5 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 D.6 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 D.7 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Bibliography 297 Index 299