Problems and Solutions for Undergraduate Real Analysis

The present book Problems and Solutions for Undergraduate Real Analysis is the combined volume of author’s two books Problems and Solutions for Undergraduate Real Analysis I and Problems and Solutions for Undergraduate Real Analysis II . By offering 456 exercises with different levels of difficulty, this book gives a brief exposition of the foundations of first-year undergraduate real analysis. Furthermore, we believe that students and instructors may find that the book can also be served as a source for some advanced courses or as a reference.The wide variety of problems, which are of varying difficulty, include the following topics: Elementary Set Algebra The Real Number System Countable and Uncountable Sets Elementary Topology on Metric Spaces Sequences in Metric Spaces Series of Numbers Limits and Continuity of Functions Differentiation The Riemann-Stieltjes Integral Sequences and Series of Functions Improper Integrals Lebesgue Measure Lebesgue Measurable Functions Lebesgue Integration Differential Calculus of Functions of Several Variables Integral Calculus of Functions of Several Variables Furthermore, the main features of this book are listed as follows: The book contains 456 problems of undergraduate real analysis, which cover the topics mentioned above, with detailed and complete solutions. In fact, the solutions show every detail, every step and every theorem that I applied. Each chapter starts with a brief and concise note of introducing the notations, terminologies, basic mathematical concepts or important/famous/frequently used theorems (without proofs) relevant to the topic. As a consequence, students can use these notes as a quick review before midterms or examinations. Three levels of difficulty have been assigned to problems so that you can sharpen your mathematics step-by-step. Different colors are used frequently in order to highlight or explain problems, examples, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. (ebook only) An appendix about mathematical logic is included. It tells students what concepts of logic (e.g. techniques of proofs) are necessary in advanced mathematics. Preface 5 List of Figures 7 List of Tables 9 Elementary Set Algebra 15 Fundamental Concepts 15 Sets, Functions and Relations 18 Mathematical Induction 20 The Real Number System 23 Fundamental Concepts 23 Rational and Irrational Numbers 24 Absolute Values 26 The Completeness Axiom 27 Countable and Uncountable Sets 33 Fundamental Concepts 33 Problems on Countable and Uncountable Sets 34 Elementary Topology on Metric Spaces 41 Fundamental Concepts 41 Open Sets and Closed Sets 45 Compact Sets 52 The Heine-Borel Theorem 57 Connected Sets 59 Sequences in Metric Spaces 63 Fundamental Concepts 63 Convergence of Sequences 66 Upper and Lower Limits 73 Cauchy Sequences and Complete Metric Spaces 79 Recurrence Relations 84 Series of Numbers 89 Fundamental Concepts 89 Convergence of Series of Nonnegative Terms 93 Alternating Series and Absolute Convergence 101 The Series n=1anbn and Multiplication of Series 104 Power Series 107 Limits and Continuity of Functions 111 Fundamental Concepts 111 Limits of Functions 116 Continuity and Uniform Continuity of Functions 122 The Extreme Value Theorem and the Intermediate Value Theorem 130 Discontinuity of Functions 134 Monotonic Functions 136 Differentiation 141 Fundamental Concepts 141 Properties of Derivatives 146 The Mean Value Theorem for Derivatives 152 L'Hôspital's Rule 160 Higher Order Derivatives and Taylor's Theorem 163 Convexity and Derivatives 167 The Riemann-Stieltjes Integral 171 Fundamental Concepts 171 Integrability of Real Functions 176 Applications of Integration Theorems 183 The Mean Value Theorems for Integrals 195 Sequences and Series of Functions 201 Fundamental Concepts 201 Uniform Convergence for Sequences of Functions 206 Uniform Convergence for Series of Functions 215 Equicontinuous Families of Functions 222 Approximation by Polynomials 227 Improper Integrals 233 Fundamental Concepts 233 Evaluations of Improper Integrals 237 Convergence of Improper Integrals 242 Miscellaneous Problems on Improper Integrals 250 Lebesgue Measure 257 Fundamental Concepts 257 Lebesgue Outer Measure 260 Lebesgue Measurable Sets 265 Necessary and Sufficient Conditions for Measurable Sets 277 Lebesgue Measurable Functions 283 Fundamental Concepts 283 Lebesgue Measurable Functions 285 Applications of Littlewood's Three Principles 296 Lebesgue Integration 305 Fundamental Concepts 305 Properties of Integrable Functions 309 Applications of Fatou's Lemma 322 Applications of Convergence Theorems 329 Differential Calculus of Functions of Several Variables 341 Fundamental Concepts 341 Differentiation of Functions of Several Variables 346 The Mean Value Theorem for Differentiable Functions 355 The Inverse Function Theorem and the Implicit Function Theorem 357 Higher Order Derivatives 361 Integral Calculus of Functions of Several Variables 367 Fundamental Concepts 367 Jordan Measurable Sets 374 Integration on Rn 378 Applications of the Mean Value Theorem 386 Applications of the Change of Variables Theorem 389 Appendix 397 Language of Mathematics 397 Fundamental Concepts 397 Statements and Logical Connectives 398 Quantifiers and their Basic Properties 400 Necessity and Sufficiency 401 Techniques of Proofs 402 Index 405 Bibliography 411