Concise introduction to basic real analysis

This book provides an introduction to basic topics in Real Analysis and makes the subject easily understandable to all learners. The book is useful for those that are involved with Real Analysis in disciplines such as mathematics, engineering, technology, and other physical sciences. It provides a good balance while dealing with the basic and essential topics that enable the reader to learn the more advanced topics easily. It includes many examples and end of chapter exercises including hints for solutions in several critical cases. The book is ideal for students, instructors, as well as those doing research in areas requiring a basic knowledge of Real Analysis. Those more advanced in the field will also find the book useful to refresh their knowledge of the topic. Features Includes basic and essential topics of real analysis Adopts a reasonable approach to make the subject easier to learn Contains many solved examples and exercise at the end of each chapter Presents a quick review of the fundamentals of set theory Covers the real number system Discusses the basic concepts of metric spaces and complete metric spaces;Review of set theory -- The real number system -- Sequences and series of real numbers -- Metric spaces: basic concepts, complete metric spaces -- Limits and continuity -- Connectedness and compactness -- Differentiation -- Integration -- Sequences and series of functions Cover......Page 1 Half Title......Page 2 Title Page......Page 4 Copyright Page......Page 5 Contents......Page 6 Preface......Page 10 Authors......Page 12 1.1 Introduction and Notations......Page 14 1.3 Relations and Functions......Page 15 1.4 Countable and Uncountable Sets......Page 17 1.5 Set Algebras......Page 19 1.6 Exercises......Page 22 2.1 Field Axioms......Page 26 2.2 Order Axioms......Page 27 2.3 Geometrical Representation of Real Numbers and Intervals......Page 28 2.5 Upper Bounds, Least Upper Bound or Supremum, the Completeness Axiom, Archimedean Property of......Page 29 2.6 Infinite Decimal Representation of Real Numbers......Page 31 2.7 Absolute Value, Triangle Inequality, Cauchy-Schwarz Inequality......Page 33 2.8 Extended Real Number System R*......Page 36 2.9 Exercises......Page 37 3.1 Convergent and Divergent Sequences of Real Numbers......Page 38 3.2 Limit Superior and Limit Inferior of a Sequence of Real Numbers......Page 39 3.3 Infinite Series of Real Numbers......Page 41 3.4 Convergence Tests for Infinite Series......Page 47 3.5 Rearrangements of Series......Page 50 3.6 Riemann's Theorem on Conditionally Convergent Series of Real Numbers......Page 51 3.7 Cauchy Multiplications of Series......Page 52 3.8 Exercises......Page 54 4.1 Metric and Metric Spaces......Page 58 4.2 Point Set Topology in Metric Spaces......Page 59 4.3 Convergent and Divergent Sequences in a Metric Space......Page 66 4.4 Cauchy Sequences and Complete Metric Spaces......Page 67 4.5 Exercises......Page 69 5.1 The Limit of Functions......Page 74 5.2 Algebras of Limits......Page 76 5.3 Right-Hand and Left-Hand Limits......Page 79 5.4 Infinite Limits and Limits at Infinity......Page 82 5.5 Certain Important Limits......Page 83 5.6 Sequential Definition of Limit of a Function......Page 84 5.7 Cauchy's Criterion for Finite Limits......Page 85 5.8 Monotonic Functions......Page 86 5.10 Continuous and Discontinuous Functions......Page 88 5.11 Some Theorems on the Continuity......Page 93 5.12 Properties of Continuous Functions......Page 96 5.13 Uniform Continuity......Page 98 5.14 Continuity and Uniform Continuity in Metric Spaces......Page 101 5.15 Exercises......Page 104 6.1 Connectedness......Page 112 6.2 The Intermediate Value Theorem......Page 118 6.3 Components......Page 120 6.4 Compactness......Page 121 6.5 The Finite Intersection Property......Page 127 6.6 The Heine-Borel Theorem......Page 129 6.7 Exercises......Page 133 7.1 The Derivative......Page 136 7.2 The Differential Calculus......Page 139 7.3 Properties of Differentiable Functions......Page 145 7.4 The L'Hospital Rule......Page 151 7.5 Taylor's Theorem......Page 160 7.6 Exercises......Page 167 8.1 The Riemann Integral......Page 170 8.2 Properties of the Riemann Integral......Page 181 8.3 The Fundamental Theorems of Calculus......Page 187 8.4 The Substitution Theorem and Integration by Parts......Page 192 8.5 Improper Integrals......Page 194 8.6 The Riemann-Stieltjes Integral......Page 200 8.7 Functions of Bounded Variation......Page 209 8.8 Exercises......Page 218 9.1 The Pointwise Convergence of Sequences of Functions and the Uniform Convergence......Page 226 9.2 The Uniform Convergence and the Continuity, the Cauchy Criterion for the Uniform Convergence......Page 228 9.3 The Uniform Convergence of Infinite Series of Functions......Page 230 9.4 The Uniform Convergence of Integrations and Differentiations......Page 232 9.5 The Equicontinuous Family of Functions and the Arzela-Ascoli Theorem......Page 235 9.6 Dirichlet's Test for the Uniform Convergence......Page 237 9.7 The Weierstrass Theorem......Page 238 9.8 Some Examples......Page 240 9.9 Exercises......Page 245 Bibliography......Page 248 Index......Page 250 Cover -- Half Title -- Title Page -- Copyright Page -- Contents -- Preface -- Authors -- 1 Review of Set Theory -- 1.1 Introduction and Notations -- 1.2 Ordered Pairs and Cartesian Product -- 1.3 Relations and Functions -- 1.4 Countable and Uncountable Sets -- 1.5 Set Algebras -- 1.6 Exercises -- 2 The Real Number System -- 2.1 Field Axioms -- 2.2 Order Axioms -- 2.3 Geometrical Representation of Real Numbers and Intervals -- 2.4 Integers, Rational Numbers, and Irrational Numbers -- 2.5 Upper Bounds, Least Upper Bound or Supremum, the Completeness Axiom, Archimedean Property of -- 2.6 Infinite Decimal Representation of Real Numbers -- 2.7 Absolute Value, Triangle Inequality, Cauchy-Schwarz Inequality -- 2.8 Extended Real Number System R* -- 2.9 Exercises -- 3 Sequences and Series of Real Numbers -- 3.1 Convergent and Divergent Sequences of Real Numbers -- 3.2 Limit Superior and Limit Inferior of a Sequence of Real Numbers -- 3.3 Infinite Series of Real Numbers -- 3.4 Convergence Tests for Infinite Series -- 3.5 Rearrangements of Series -- 3.6 Riemann's Theorem on Conditionally Convergent Series of Real Numbers -- 3.7 Cauchy Multiplications of Series -- 3.8 Exercises -- 4 Metric Spaces - Basic Concepts, Complete Metric Spaces -- 4.1 Metric and Metric Spaces -- 4.2 Point Set Topology in Metric Spaces -- 4.3 Convergent and Divergent Sequences in a Metric Space -- 4.4 Cauchy Sequences and Complete Metric Spaces -- 4.5 Exercises -- 5 Limits and Continuity -- 5.1 The Limit of Functions -- 5.2 Algebras of Limits -- 5.3 Right-Hand and Left-Hand Limits -- 5.4 Infinite Limits and Limits at Infinity -- 5.5 Certain Important Limits -- 5.6 Sequential Definition of Limit of a Function -- 5.7 Cauchy's Criterion for Finite Limits -- 5.8 Monotonic Functions -- 5.9 The Four Functional Limits at a Point -- 5.10 Continuous and Discontinuous Functions "This book provides an introduction to basic topics in Real Analysis and makes the subject easily understandable to all learners. The book is useful for those that are involved with Real Analysis in disciplines such as mathematics, engineering, technology, and other physical sciences. It provides a good balance while dealing with the basic and essential topics that enable the reader to learn the more advanced topics easily. It includes many examples and end of chapter exercises including hints for solutions in several critical cases. The book is ... for students, instructors, as well as those doing research in areas requiring a basic knowledge of Real Analysis. Those more advanced in the field will also find the book useful to refresh their knowledge of the topic."--Provided by publisher This concise book includes all basic and essential topics of Real Analysis, for the first time learner. It is for those that are involved with Real Analysis in disciplines such as engineering, technology, and other physical sciences.