A Course in Real Analysis

'A Course in Real Analysis' provides a rigorous treatment of the foundations of differential and integral calculus at the advanced undergraduate level. The book's material has been extensively classroom tested in the author's two-semester undergraduate course on real analysis at the George Washington University. A Course in Real Analysis provides a rigorous treatment of the foundations of differential and integral calculus at the advanced undergraduate level. The book's material has been extensively classroom tested in the author's two-semester undergraduate course on real analysis at The George Washington University. The first part of the text presents the calculus of functions of one variable. This part covers traditional topics, such as sequences, continuity, differentiability, Riemann integrability, numerical series, and the convergence of sequences and series of functions. It also includes optional sections on Stirling's formula, functions of bounded variation, Riemann-Stieltjes integration, and other topics. The second part focuses on functions of several variables. It introduces the topological ideas (such as compact and connected sets) needed to describe analytical properties of multivariable functions. This part also discusses differentiability and integrability of multivariable functions and develops the theory of differential forms on surfaces in Rn. The third part consists of appendices on set theory and linear algebra as well as solutions to some of the exercises. A full solutions manual offers complete solutions to all exercises for qualifying instructors. With clear proofs, detailed examples, and numerous exercises, this textbook gives a thorough treatment of the subject. It progresses from single variable to multivariable functions, providing a logical development of material that will prepare students for more advanced analysis-based courses Cover 1 S Title 2 A COURSE IN REAL ANALYSIS 4 © 2015 by Taylor & Francis Group, LLC 5 ISBN 978-1-4822-1928-9 (eBook - PDF) 5 Dedeication 6 Contents 8 Preface 12 List of Figures 14 List of Tables 18 List of Symbols 20 Part I: Functions of One Variable 26 Chapter 1: The Real Number System 28 1.1 From Natural Numbers to Real Numbers 28 1.2 Algebraic Properties of R 29 Exercises 32 1.3 Order Structure of R 33 Exercises 35 1.4 Completeness Property of R 37 Exercises 41 1.5 Mathematical Induction 44 Exercises 46 1.6 Euclidean Space 49 Exercises 51 Chapter 2: Numerical Sequences 54 2.1 Limits of Sequences 54 Exercises 59 2.2 Monotone Sequences 61 Exercises 62 2.3 Subsequences and Cauchy Sequences 63 Exercises 66 2.4 Limits Inferior and Superior 67 Exercises 69 Chapter 3: Limits and Continuity on R 72 3.1 Limit of a Function 72 Exercises 79 *3.2 Limits Inferior and Superior 80 Exercises 83 3.3 Continuous Functions 84 Exercises 86 3.4 Properties of Continuous Functions 88 Exercises 90 3.5 Uniform Continuity 92 Exercises 95 Chapter 4: Differentiation on R 98 4.1 Definition of Derivative and Examples 98 Exercises 102 4.2 The Mean Value Theorem 105 Exercises 107 *4.3 Convex Functions 110 4.4 Inverse Functions 113 Exercises 117 4.5 L’Hospital’s Rule 119 Exercises 123 4.6 Taylor’s Theorem on R 125 Exercises 127 *4.7 Newton’s Method 128 Exercises 130 Chapter 5: Riemann Integration on R 132 5.1 The Riemann–Darboux Integral 132 Exercises 140 5.2 Properties of the Integral 141 Exercises 144 5.3 Evaluation of the Integral 145 Exercises 151 *5.4 Stirling’s Formula 154 5.5 Integral Mean Value Theorems 156 Exercises 158 *5.6 Estimation of the Integral 159 5.7 Improper Integrals 168 Exercises 174 5.8 A Deeper Look at Riemann Integrability 176 Exercises 177 *5.9 Functions of Bounded Variation 177 *5.10 The Riemann–Stieltjes Integral 181 Chapter 6: Numerical Infinite Series 188 6.1 Definition and Examples 188 Exercises 191 6.2 Series with Nonnegative Terms 194 Exercises 198 6.3 More Refined Convergence Tests 201 Exercises 204 6.4 Absolute and Conditional Convergence 206 Exercises 211 *6.5 Double Sequences and Series 213 Exercises 216 Chapter 7: Sequences and Series of Functions 218 7.1 Convergence of Sequences of Functions 218 Exercises 222 7.2 Properties of the Limit Function 224 Exercises 227 7.3 Convergence of Series of Functions 229 Exercises 233 7.4 Power Series 236 Exercises 248 Part II: Functions of Several Variables 254 Chapter 8: Metric Spaces 256 8.1 Definitions and Examples 256 Exercises 260 8.2 Open and Closed Sets 263 Exercises 267 8.3 Closure, Interior, and Boundary 268 Exercises 271 8.4 Limits and Continuity 273 Exercises 278 8.5 Compact Sets 280 Exercises 285 *8.6 The Arzelà–Ascoli Theorem 288 Exercises 291 8.7 Connected Sets 293 Exercises 298 8.8 The Stone–Weierstrass Theorem 300 Exercises 305 *8.9 Baire’s Theorem 307 Exercises 311 Chapter 9: Differentiation on R^n 312 9.1 Definition of the Derivative 312 Exercises 318 9.2 Properties of the Differential 320 Exercises 324 9.3 Further Properties of the Differential 326 Exercises 329 9.4 Inverse Function Theorem 331 Exercises 336 9.5 Implicit Function Theorem 337 Exercises 340 9.6 Higher Order Partial Derivatives 343 Exercises 346 9.7 Higher Order Differentials and Taylor’s Theorem 348 Exercises 353 *9.8 Optimization 355 Exercises 363 Chapter 10: Lebesgue Measure on R^n 368 10.1 General Measure Theory 368 Exercises 371 10.2 Lebesgue Outer Measure 372 Exercises 375 10.3 Lebesgue Measure 376 Exercises 381 10.4 Borel Sets 381 Exercises 384 10.5 Measurable Functions 385 Exercises 390 Chapter 11: Lebesgue Integration on R^n 392 11.1 Riemann Integration on R^n 392 11.2 The Lebesgue Integral 393 Exercises 402 11.3 Convergence Theorems 404 Exercises 407 11.4 Connections with Riemann Integration 410 11.5 Iterated Integrals 413 Exercises 420 11.6 Change of Variables 423 Exercises 432 Chapter 12: Curves and Surfaces in R^n 434 12.1 Parameterized Curves 434 Exercises 436 12.2 Integration on Curves 437 Exercises 444 12.3 Parameterized Surfaces 447 Exercises 456 12.4 m-Dimensional Surfaces 457 Exercises 468 Chapter 13: Integration on Surfaces 472 13.1 Differential Forms 472 Exercises 485 13.2 Integrals on Parameterized Surfaces 486 Exercises 494 13.3 Partitions of Unity 497 13.4 Integration on Compact m-Surfaces 500 Exercises 502 13.5 The Fundamental Theorems of Calculus 503 Exercises 516 *13.6 Closed Forms in R^n 520 Part III: Appendices 528 Appendix A: Set Theory 530 Appendix B: Linear Algebra 534 Appendix C: Solutions to Selected Problems 542 Bibliography 606 Back Cover 608 Functions of One Variable The Real Number System From Natural Numbers to Real Numbers Algebraic Properties of R Order Structure of R Completeness Property of R Mathematical Induction Euclidean Space Numerical Sequences Limits of Sequences Monotone Sequences Subsequences. Cauchy Sequences Limit Inferior and Limit Superior Limits and Continuity on R Limit of a Function Limits Inferior and Superior Continuous Functions Some Properties of Continuous Functions Uniform Continuity Differentiation on R Definition of Derivative. Examples The Mean Value Theorem Convex Functions Inverse Functions L'Hospital's Rule Taylor's Theorem on R Newton's Method Riemann Integration on R The Riemann-Darboux Integral Properties of the Integral Evaluation of the Integral Stirling's Formula Integral Mean Value Theorems Estimation of the Integral Improper Integrals A Deeper Look at Riemann Integrability Functions of Bounded Variation The Riemann-Stieltjes Integral Numerical Infinite Series Definition and Examples Series with Nonnegative Terms More Refined Convergence Tests Absolute and Conditional Convergence Double Sequences and Series Sequences and Series of Functions Convergence of Sequences of Functions Properties of the Limit Function Convergence of Series of Functions Power Series Functions of Several Variables Metric Spaces Definitions and Examples Open and Closed Sets Closure, Interior, and Boundary Limits and Continuity Compact Sets The Arzelà-Ascoli Theorem Connected Sets The Stone-Weierstrass Theorem Baire's Theorem Differentiation on RnDefinition of the Derivative Properties of the Differential Further Properties of the Derivative The Inverse Function Theorem The Implicit Function Theorem Higher Order Partial DerivativesHigher Order Differentials. Taylor's Theorem on Rn Optimization Lebesgue Measure on Rn Some General Measure Theory Lebesgue Outer Measure Front Cover; Contents; Preface; List of Figures; List of Tables; List of Symbols; Part I: Functions of One Variable; Chapter 1: The Real Number System; Chapter 2: Numerical Sequences; Chapter 3: Limits and Continuity on R; Chapter 4: Differentiation on R; Chapter 5: Riemann Integration on R; Chapter 6: Numerical Infinite Series; Chapter 7: Sequences and Series of Functions; Part II: Functions of Several Variables; Chapter 8: Metric Spaces; Chapter 9: Differentiation on Rn; Chapter 10: Lebesgue Measure on Rn; Chapter 11: Lebesgue Integration on Rn; Chapter 12: Curves and Surfaces in Rn