Introduction to real analysis

using An Extremely Clear And Informal Approach, This Book Introduces Readers To A Rigorous Understanding Of Mathematical Analysis And Presents Challenging Math Concepts As Clearly As Possible. The Real Number System. Differential Calculus Of Functions Of One Variable. Riemann Integral Functions Of One Variable. Integral Calculus Of Real-valued Functions. Metric Spaces. For Those Who Want To Gain An Understanding Of Mathematical Analysis And Challenging Mathematical Concepts. Contents......Page 5 Preface......Page 7 1.1 THE REAL NUMBER SYSTEM......Page 11 1.2 MATHEMATICAL INDUCTION......Page 20 1.3 THE REAL LINE......Page 29 2.1 FUNCTIONS AND LIMITS......Page 40 2.2 CONTINUITY......Page 63 2.3 DIFFERENTIABLE FUNCTIONS OF ONE VARIABLE......Page 83 2.4 L’HOSPITAL’S RULE......Page 98 2.5 TAYLOR’S THEOREM......Page 108 3 Integral Calculus of Functions of One Variable......Page 123 3.1 DEFINITION OF THE INTEGRAL......Page 124 3.2 EXISTENCE OF THE INTEGRAL......Page 138 3.3 PROPERTIES OF THE INTEGRAL......Page 145 3.4 IMPROPER INTEGRALS......Page 161 3.5 A MORE ADVANCED LOOK AT THE EXISTENCE OF THE PROPER RIEMANN INTEGRAL......Page 181 4 In nite Sequences and Series......Page 188 4.1 SEQUENCES OF REAL NUMBERS......Page 189 4.2 EARLIER TOPICS REVISITED WITH SEQUENCES......Page 205 4.3 INFINITE SERIES OF CONSTANTS......Page 210 4.4 SEQUENCES AND SERIES OF FUNCTIONS......Page 244 4.5 POWER SERIES......Page 268 5.1 STRUCTURE OF R^n......Page 292 5.2 CONTINUOUS REAL-VALUED FUNCTIONS OF n VARIABLES......Page 313 5.3 PARTIAL DERIVATIVES AND THE DIFFERENTIAL......Page 327 5.4 THE CHAIN RULE AND TAYLOR’S THEOREM......Page 350 6.1 LINEAR TRANSFORMATIONS AND MATRICES......Page 372 6.2 CONTINUITY AND DIFFERENTIABILITY OF TRANSFORMATIONS......Page 389 6.3 THE INVERSE FUNCTION THEOREM......Page 405 6.4 THE IMPLICIT FUNCTION THEOREM......Page 428 7.1 DEFINITION AND EXISTENCE OF THE MULTIPLE INTEGRAL......Page 446 7.2 ITERATED INTEGRALS AND MULTIPLE INTEGRALS......Page 473 7.3 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS......Page 495 8.1 INTRODUCTION TO METRIC SPACES......Page 529 8.2 COMPACT SETS IN A METRIC SPACE......Page 546 8.3 CONTINUOUS FUNCTIONS ON METRIC SPACES......Page 554 Answers to Selected Exercises......Page 560 Index......Page 574 Prentice Hall Contents 5 Preface 7 1 The Real Numbers 11 1.1 THE REAL NUMBER SYSTEM 11 1.2 MATHEMATICAL INDUCTION 20 1.3 THE REAL LINE 29 2 Differential Calculus of Functions of One Variable 40 2.1 FUNCTIONS AND LIMITS 40 2.2 CONTINUITY 63 2.3 DIFFERENTIABLE FUNCTIONS OF ONE VARIABLE 83 2.4 L’HOSPITAL’S RULE 98 2.5 TAYLOR’S THEOREM 108 3 Integral Calculus of Functions of One Variable 123 3.1 DEFINITION OF THE INTEGRAL 124 3.2 EXISTENCE OF THE INTEGRAL 138 3.3 PROPERTIES OF THE INTEGRAL 145 3.4 IMPROPER INTEGRALS 161 3.5 A MORE ADVANCED LOOK AT THE EXISTENCE OF THE PROPER RIEMANN INTEGRAL 181 4 In nite Sequences and Series 188 4.1 SEQUENCES OF REAL NUMBERS 189 4.2 EARLIER TOPICS REVISITED WITH SEQUENCES 205 4.3 INFINITE SERIES OF CONSTANTS 210 4.4 SEQUENCES AND SERIES OF FUNCTIONS 244 4.5 POWER SERIES 268 5 Real-Valued Functions of Several Variables 292 5.1 STRUCTURE OF R^n 292 5.2 CONTINUOUS REAL-VALUED FUNCTIONS OF n VARIABLES 313 5.3 PARTIAL DERIVATIVES AND THE DIFFERENTIAL 327 5.4 THE CHAIN RULE AND TAYLOR’S THEOREM 350 6 Vector-Valued Functions of Several Variables 372 6.1 LINEAR TRANSFORMATIONS AND MATRICES 372 6.2 CONTINUITY AND DIFFERENTIABILITY OF TRANSFORMATIONS 389 6.3 THE INVERSE FUNCTION THEOREM 405 6.4 THE IMPLICIT FUNCTION THEOREM 428 7 Integrals of Functions of Several Variables 446 7.1 DEFINITION AND EXISTENCE OF THE MULTIPLE INTEGRAL 446 7.2 ITERATED INTEGRALS AND MULTIPLE INTEGRALS 473 7.3 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS 495 8 Metric Spaces 529 8.1 INTRODUCTION TO METRIC SPACES 529 8.2 COMPACT SETS IN A METRIC SPACE 546 8.3 CONTINUOUS FUNCTIONS ON METRIC SPACES 554 Answers to Selected Exercises 560 Index 574 0130457868,9780130457868 This is a text for a two-term course in introductory real analysis for junior or senior mathematics majors and science students with a serious interest in mathematics. Prospective educators or mathematically gifted high school students can also benefit from the mathematical maturity that can be gained from an introductory real analysis course. The book is designed to fill the gaps left in the development of calculus as it is usually presented in an elementary course, and to provide the background required for insight into more advanced courses in pure and applied mathematics. The standard elementary calculus sequence is the only specific prerequisite for Chapters 1-5, which deal with real-valued functions. (However, other analysis oriented courses, such as elementary differential equation, also provide useful preparatory experience.) Chapters 6 and 7 require a working knowledge of determinants, matrices and linear transformations, typically available from a first course in linear algebra. Chapter 8 is accessible after completion of Chapters 1-5. Annotation Introducing students to an understanding of mathematical analysis, this text aims to present mathematical concepts clearly. It also provides students with the background required for insight into more advanced courses in pure and applied mathematics. It is intended for two-semester, junior/senior-level courses in introductory real analysis