Yet Another Introduction to Analysis

Citation

Bryant, V. (1990). Yet Another Introduction to Analysis. Cambridge University Press.

Chapter Summary

The book is a first course in real analysis for new university students, designed to grow out of the calculus they already know from school. The preface says that readers of the author's earlier book on metric spaces, a second-year text, asked him to recommend a similar first-year book, and this is his answer. It is organised as five chapters plus a long section of solutions to exercises, and it is written in a conversational first-person voice. The table of contents announces each chapter with a short list of questions it sets out to answer.

Chapter 1: Firm Foundations

The questions are why to start with numbers, how far primary-school arithmetic can take us and what extra assumption is needed. The chapter opens by computing derivatives and integrals of x², |x| and the integer-part function, and uses them to argue that a study of functions must begin with a study of numbers. It takes primary-school arithmetic of the integers and rationals as given, proves that no rational number has square 2, and then introduces the completeness axiom (the "piggy-in-the-middle" property) to obtain the square root of 2. Suprema, bounded sets, the maximum and minimum of sets of integers and the principle of mathematical induction follow, and the chapter ends with the numbers e and π, including a proof that e is irrational. Exercises are set between sections, with solutions at the back of the book.

Chapter 2: Gradually Getting There

The questions are why sequences are needed, how they behave and how they help to work out infinite addition sums. The chapter runs from page 24 to page 74.

Chapter 3: A Functional Approach

The questions are what the familiar functions mean, how they behave and how they are related. At about seventy pages it is the longest chapter, and its title and questions point to a systematic study of the familiar functions.

Chapter 4: Calculus at Last

The questions are how to work out gradients, how that leads to differentiation and how differentiation helps to find averages and approximations. The title suggests that differentiation arrives only after the groundwork of the earlier chapters.

Chapter 5: An Integrated Conclusion

The questions are how to define the area under a graph and why it should have anything to do with differentiation. This is the book's treatment of integration and the link between integration and differentiation.

Solutions to Exercises

About seventy pages, roughly a quarter of the book, are devoted to solutions to the exercises.

Key Concepts

Chapter 1: Firm Foundations

• Irrational number: the book shows there is no rational x with x² = 2, so that the rationals have gaps.
• Completeness axiom: if two non-empty sets L and M satisfy l ≤ m for all l in L and m in M, then some real number lies between them.
• Supremum: the least upper bound of a bounded set.
• Principle of mathematical induction: derived in the chapter from the fact that every non-empty set of positive integers has a smallest member.

Chapters 2–5

• Sequences and infinite sums, functions, gradients and differentiation, and area under a graph are the objects named by the chapter titles and opening questions.

Critical Analysis

Strengths

• Motivation before definitions. The publisher states that each concept is introduced only after its need is clear and after it has been used informally, and that new ideas are related to school topics. The opening of Chapter 1 follows this: three small calculus computations lead to the need for a precise picture of the real numbers.
• A reader who is expected to take part. Exercises are placed within chapters, and the text asks readers to think about them before looking at the solutions, which occupy a substantial section at the back. The publisher says the reader is led through each step so that they can soon predict the next one.
• Clear about its starting point. Chapter 1 states plainly that it assumes primary-school arithmetic and then adds one further assumption, completeness, and later results rest on those assumptions.
• Aimed at first-year students. The preface places the book as a first-year text, for students who will arrive with different expectations after school mathematics reform, and the author's earlier metric spaces book is said to be for second-year students.

Limitations

• Real numbers are assumed, not constructed. The book works from the stated arithmetic and the completeness axiom; readers who want a construction of the real numbers will need another source.
• Single-variable, no topology. The contents list no chapter on metric spaces or on functions of several variables; the preface mentions the author's separate metric-spaces book for second-year students.
• Hard to navigate by topic. The contents page lists five chapters with guiding questions rather than section headings, so finding a specific theorem depends on the index.
• Written for its time. The preface's argument rests on the British school reforms and the GCSE examination of 1990, so its picture of what first-year students know may not match every reader.

Where It Fits in the Course

Bryant's distinctive feature is that each idea is motivated from school calculus. Lara Alcock's How to Think About Analysis is a study guide that explains how to read definitions and proofs, and works well alongside any text. R. P. Burn's Numbers and Functions: Steps into Analysis is built from more than eight hundred questions that the reader works through, a more active format than Bryant's. Stephen Abbott's Understanding Analysis is an undergraduate text for a first rigorous course with a motivating discussion at the start of each chapter; it adds chapters on the basic topology of the real line and on sequences and series of functions, and is a natural next step. Bartle and Sherbert's Introduction to Real Analysis covers sequences, limits, continuity, differentiation, the Riemann integral and sequences of functions in order, and Rudin's Principles of Mathematical Analysis is a concise, rigorous text built on metric spaces that extends to several variables. Both are more formal than Bryant.

How to Use This Book

The book is for students at the start of a first analysis course who have studied school calculus. Little else is needed beyond willingness to work through questions.

Start with Chapter 1 and read it carefully, including the exercises: it sets the assumptions used in the rest of the book and introduces the completeness axiom and induction. Chapters 2 to 5 follow in order, since the titles describe one continuous argument from sequences through functions and differentiation to integration.

Attempt each exercise before opening the solutions section at the back, and use it to check your reasoning. After finishing, move to a more formal text such as Abbott, Bartle and Sherbert or Rudin, which develops topology and sequences of functions more fully.