Undergraduate Analysis: A Working Textbook

Description

Analysis underpins calculus, much as calculus underpins virtually all mathematical sciences. A sound understanding of analysis' results and techniques is therefore valuable for a wide range of disciplines both within mathematics itself and beyond its traditional boundaries. This text seeks to develop such an understanding for undergraduate students on mathematics and mathematically related programmes. Keenly aware of contemporary students' diversity of motivation, background knowledge and time pressures, it consistently strives to blend beneficial aspects of the workbook, the formal teaching text, and the informal and intuitive tutorial discussion. The authors devote ample space and time for development of confidence in handling the fundamental ideas of the topic. They also focus on learning through doing, presenting a comprehensive range of examples and exercises, some worked through in full detail, some supported by sketch solutions and hints, some left open to the reader's initiative. Without undervaluing the absolute necessity of secure logical argument, they legitimise the use of informal, heuristic, even imprecise initial explorations of problems aimed at deciding how to tackle them. In this respect they authors create an atmosphere like that of an apprenticeship, in which the trainee analyst can look over the shoulder of the experienced practitioner. Cover Undergraduate Analysis: A Working Textbook Copyright Dedication Preface Contents A Note to the Instructor A Note to the Student Reader 1 Preliminaries 1.1 Real numbers 1.2 The basic rules of inequalities — a checklist of things you probably know already 1.3 Modulus 1.4 Floor 2 Limit of a sequence — an idea, a definition, a tool 2.1 Introduction 2.2 Sequences, and how to write them 2.3 Approximation 2.4 Infinite decimals 2.5 Approximating an area 2.6 A small slice of π 2.7 Testing limits by the definition 2.8 Combining sequences; the algebra of limits 2.9 POSTSCRIPT: to infinity 2.10 Important note on ‘elementary functions’ 3 Interlude: different kinds of numbers 3.1 Sets 3.2 Intervals, max and min, sup and inf 3.3 Denseness 4 Up and down — increasing and decreasing sequences 4.1 Monotonic bounded sequences must converge 4.2 Induction: infinite returns for finite effort 4.3 Recursively defined sequences 4.4 POSTSCRIPT: The epsilontics game — the ‘fifth factor of difficulty’ 5 Sampling a sequence — subsequences 5.1 Introduction 5.2 Subsequences 5.3 Bolzano-Weierstrass: the overcrowded interval 6 Special (or specially awkward) examples 6.1 Introduction 6.2 Important examples of convergence 7 Endless sums — a first look at series 7.1 Introduction 7.2 Definition and easy results 7.3 Big series, small series: comparison tests 7.4 The root test and the ratio test 8 Continuous functions — the domain thinks that the graph is unbroken 8.1 Introduction 8.2 An informal view of continuity 8.3 Continuity at a point 8.4 Continuity on a set 8.5 Key theorems on continuity 8.6 Continuity of the inverse 9 Limit of a function 9.1 Introduction 9.2 Limit of a function at a point 10 Epsilontics and functions 10.1 The epsilontic view of function limits 10.2 The epsilontic view of continuity 10.3 One-sided limits 11 Infinity and function limits 11.1 Limit of a function as x tends to infinity or minus infinity 11.2 Functions tending to infinity or minus infinity 12 Differentiation — the slope of the graph 12.1 Introduction 12.2 The derivative 12.3 Up and down, maximum and minimum: for differentiable functions 12.4 Higher derivatives 12.5 Alternative proof of the chain rule 13 The Cauchy condition — sequences whose terms pack tightly together 13.1 Cauchy equals convergent 14 More about series 14.1 Absolute convergence 14.2 The ‘robustness’ of absolutely convergent series 14.3 Power series 15 Uniform continuity — continuity’s global cousin 15.1 Introduction 15.2 Uniformly continuous functions 15.3 The bounded derivative test 16 Differentiation — mean value theorems, power series 16.1 Introduction 16.2 Cauchy and l’Hôpital 16.3 Taylor series 16.4 Differentiating a power series 17 Riemann integration — area under a graph 17.1 Introduction 17.2 Riemann integrability — how closely can rectangles approximate areas under graphs? 17.3 The integral theorems we ought to expect 17.4 The fundamental theorem of calculus 18 The elementary functions revisited 18.1 Introduction 18.2 Logarithms and exponentials 18.3 Trigonometric functions 19 Exercises: for additional practice Suggestions for further reading Index