Real Analysis

Real Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates. Combining an informal style with precision mathematics, the book covers all the key topics with fully worked examples and exercises with solutions. All the concepts and techniques are deployed in examples in the final chapter to provide the student with a thorough understanding of this challenging subject. This book offers a fresh approach to a core subject and manages to provide a gentle and clear introduction without sacrificing rigour or accuracy. Preface.................................................................... 6 Contents................................................................... 8 1. Introductory Ideas...................................................... 10 1.1 Foreword for the Student: Is Analysis Necessary.................... 10 1.2 The Concept of Number.............................................. 12 1.3 The Language of Set Theory......................................... 13 1.4 Real Numbers....................................................... 16 1.5 Induction.......................................................... 21 1.6 Inequalities....................................................... 27 2. Sequences and Series.................................................... 36 2.1 Sequences.......................................................... 36 2.2 Sums, Products and Quotients....................................... 42 2.3 Monotonic Sequences................................................ 46 2.4 Cauchy Sequences................................................... 51 2.5 Series............................................................. 56 2.6 The Comparison Test................................................ 59 2.7 Series of Positive and Negative Terms.............................. 67 3. Functions and Continuity................................................ 72 3.1 Functions, Graphs.................................................. 72 3.2 Sums, Products, Compositions; Polynomial and Rational Functions.... 75 3.3 Circular Functions................................................. 79 3.4 Limits............................................................. 82 3.5 Continuity......................................................... 90 3.6 Uniform Continuity................................................. 99 3.7 Inverse Functions..................................................103 4. Differentiation.........................................................108 4.1 The Derivative.....................................................108 4.2 The Mean Value Theorems............................................114 4.3 Inverse Functions..................................................119 4.4 Higher Derivatives.................................................122 4.5 Taylor's Theorem...................................................125 5. Integration.............................................................128 5.1 The Riemann Integral...............................................128 5.2 Classes of Integrable Functions....................................135 5.3 Properties of Integrals............................................140 5.4 The Fundamental Theorem............................................147 5.5 Techniques of Integration..........................................152 5.6 Improper Integrals of the First Kind...............................159 5.7 Improper Integrals of the Second Kind..............................167 6. The Logarithmic and Exponential Functions...............................174 6.1 A Function Defined by an Integral..................................174 6.2 The Inverse Function...............................................177 6.3 Further Properties of the Exponential and Logarithmic Functions....185 7. Sequences and Series of Functions.......................................190 7.1 Uniform Convergence................................................190 7.2 Uniform Convergence of Series......................................201 7.3 Power Series.......................................................210 8. The Circular Functions..................................................226 8.1 Definitions and Elementary Properties..............................226 8.2 Lengt h............................................................229 9. Miscellaneous Examples..................................................238 9.1 Wallis's Formula...................................................238 9.2 Stirling's Formula.................................................239 9.3 A Continuous, Nowhere Differentiable Function......................243 Solutions to Exercises.....................................................246 The Greek Alphabet.........................................................278 Bibliography...............................................................280 Index......................................................................282 From the point of view of strict logic, a rigorous course on real analysis should precede a course on calculus. Strict logic, is, however, overruled by both history and practicality. Historically, calculus, with its origins in the 17th century, came first, and made rapid progress on the basis of informal intuition. Not until well through the 19th century was it possible to claim that the edifice was constructed on sound logical foundations. As for practicality, every university teacher knows that students are not ready for even a semi-rigorous course on analysis until they have acquired the intuitions and the sheer technical skills that come from a traditional calculus course. 1 Real analysis, I have always thought, is the pons asinorv.m of modern mathematics. This shows, I suppose, how much progress we have made in two thousand years, for it is a great deal more sophisticated than the Theorem of Pythagoras, which once received that title. All who have taught the subject know how patient one has to be, for the ideas take root gradually, even in students of good ability. This is not too surprising, since it took more than two centuries for calculus to evolve into what we now call analysis, and even a gifted student, guided by an expert teacher, cannot be expected to grasp all of the issues immediately. Understanding The Concepts And Methods Of Real Analysis Is An Essential Skill For Every Undergraduate Mathematics Student. Written In An Easy-to-read Style, Real Analysis Is A Comprehensive Introduction To This Core Subject And Is Ideal For Self-study Or As A Course Textbook For First And Second-year Undergraduates. Combining An Informal Style With Precision Mathematics, Real Analysis Covers All The Key Topics With Fully Worked Examples And Exercises With Solutions. Featuring: Sequences And Series - Considering The Central Notion Of A Limit.- Continuous Functions.- Differentiation.- Integration.- Logarithmic And Exponential Functions.- Uniform Convergence.- Circular Functions All These Concepts And Techniques Are Deployed In Examples In The Final Chapter To Provide The Student With A Thorough Understanding Of This Challenging Subject. Introductory Ideas -- Sequences And Series -- Functions And Continuity -- Differentiation -- Integration -- The Logarithmic And Exponential Functions -- Sequences And Series Of Functions -- The Circular Functions -- Miscellaneous Examples -- Solutions To Exercises -- Appendix: The Greek Alphabet -- Bibliography -- Index. By John M. Howie. In writing this book my assumption has been that you have encountered the fundamental ideas of analysis (function, limit, continuity, differentiation, integration) in a standard course on calculus.