How to Think About Analysis

Citation

Alcock, L. (2014). How to Think About Analysis. Oxford University Press.

Chapter Summary

The book is a study guide rather than a textbook. It has two parts: Part 1, "Studying Analysis" (Chapters 1–4), on studying Analysis in general, and Part 2, "Concepts in Analysis" (Chapters 5–10), on the central concepts, framed by an introduction and a conclusion. Every chapter in Part 2 opens with an overview and closes with a "Looking ahead" section. The introduction states that the book covers less content than an early Analysis course does, and gives much more attention to the basics and to the psychological side of learning them.

Chapter 1: What is Analysis Like?

The chapter shows a typical page of Analysis lecture notes on sequence convergence, not expecting it to be understood yet, and uses it to show what definitions, theorems and proofs look like, how symbols and words should be read, and how this kind of mathematics differs from earlier procedural work.

Chapter 2: Axioms, Definitions and Theorems

The components of a mathematical theory are taken one at a time. Sections on relating a definition to one example and then to more examples, on precision in using definitions, on examining the premises of a theorem, on diagrams and generality, and on theorems and their converses turn reading into an active skill.

Chapter 3: Proofs

This chapter explains how proofs fit into the structure of a mathematical theory and how Analysis is usually taught. It then turns to studying proofs, including a section on self-explanation, the research-based reading technique the book recommends, and ends by distinguishing reading proofs from constructing them.

Chapter 4: Learning Analysis

Practical study advice: what the Analysis experience is like, keeping up with a course, avoiding time-wasting, getting questions answered and adjusting a study strategy.

Chapter 5: Sequences

The longest chapter moves from what a sequence is and how to represent it, through monotonicity, boundedness and convergence, to the formal definition of convergence, approached both "intuition first" and "definition first". It then covers proving that a sequence converges, combining convergent sequences, and sequences that tend to infinity.

Chapter 6: Series

Series are introduced through notation and partial sums, with geometric series revisited. Two sections built around surprising examples prepare the ground for tests for convergence and alternating series; the chapter closes with power series, radius of convergence and Taylor series.

Chapter 7: Continuity

After discussing function examples and how functions are specified, including some less familiar examples, the chapter again approaches the definition of continuity both intuition first and definition first. It covers variants of the definition, proving that a function is continuous, combining continuous functions, further continuity theorems, and limits and discontinuities.

Chapter 8: Differentiability

The chapter begins with some common misconceptions about differentiability before giving the definition and applying it. Further sections cover non-differentiability, theorems involving differentiable functions and Taylor's Theorem.

Chapter 9: Integrability

Integration is rebuilt from the relationship between areas and antiderivatives, through approximating areas to a definition of integrability. The chapter presents a non-integrable function, Riemann's condition and theorems involving integrable functions, and ends with the Fundamental Theorem of Calculus.

Chapter 10: The Real Numbers

The final chapter starts from "things you don't know about numbers", treats decimal expansions, rational and irrational numbers, and the axioms for the real numbers, and ends with completeness.

Key Concepts

Chapters 1–4: Studying Analysis

• Definitions as working tools: a definition is understood by testing it against specific examples and checking each condition precisely.
• Theorem premises and converses: checking what a theorem assumes, and whether its converse holds, is part of understanding it.
• Self-explanation: explaining each line of a proof to oneself, a study technique the book bases on research in mathematics education.
• Structure of a theory: how axioms, definitions, theorems and proofs depend on one another, and how lecture courses present them.

Chapters 5–6: Sequences; Series

• Convergence of a sequence: the formal definition, read alongside the intuition it captures.
• Partial sums: a series converges when its sequence of partial sums does.
• Tests for convergence and radius of convergence: tools for deciding whether a series, and a power series in particular, converges.

Chapters 7–9: Continuity; Differentiability; Integrability

• Continuity definition and its variants: equivalent ways of stating the definition, and how to use them to prove continuity.
• Differentiability from the definition: what the definition says, where common misconceptions arise, and functions that fail to be differentiable.
• Riemann's condition: a criterion for integrability, used together with a non-integrable example.
• Fundamental Theorem of Calculus: the formal link between integration and differentiation.

Chapter 10: The Real Numbers

• Axioms for the real numbers: the properties that the real number system is assumed to satisfy.
• Completeness: the axiom that separates the real numbers from the rationals.

Critical Analysis

Strengths

• It teaches how to read Analysis. Part 1 is devoted to skills: reading definitions against examples, examining theorem premises and converses, and studying proofs through self-explanation. The author describes the advice as research-based, and every illustration is drawn from Analysis itself.
• It explains the difficulties. The introduction says the book highlights common errors, misconceptions and sources of confusion, including those that come from over-generalising school mathematics. Section titles such as "Some common misconceptions" and the paired "intuition first" and "definition first" sections show this in practice.
• It fits different course orders. Each Part 2 chapter opens with an overview so that students can match it to their own course specification. The introduction notes that some universities start with sequences and series and others with continuity, differentiability and integration.
• It looks ahead. Every Part 2 chapter ends with a "Looking ahead" section, which helps a student see how the basics connect to what a full course will cover next.

Limitations

• Deliberately limited content. The author states that early Analysis courses will cover more than the book does. It is not a replacement for a course text; topics such as compactness, uniform convergence or metric spaces do not appear in its contents.
• Short on general study skills. The study advice in Part 1 is more condensed than in the author's How to Study for a Mathematics Degree (US edition How to Study as a Mathematics Major). The introduction itself suggests starting with one of those books if undergraduate mathematics is entirely new.
• Not a fast read. The author warns that no one should expect to read the whole book quickly, and advises skipping a difficult section and coming back to it.

Where It Fits in the Course

This is the gentlest way into this course list: a companion that explains the thinking behind Analysis rather than a course text. It works well before or alongside R. P. Burn's Numbers and Functions: Steps into Analysis, which teaches the same topics through sequences of questions for the student to answer. Alcock explains how to read and build proofs, and Burn gives structured practice in doing so.

Once the habits are in place, the next step is a full course text. Stephen Abbott's Understanding Analysis is a rigorous single-variable text with motivating discussions, and Bartle and Sherbert's Introduction to Real Analysis is a more comprehensive textbook. Walter Rudin's Principles of Mathematical Analysis is the terse, formal classic, for later. Victor Bryant's Yet Another Introduction to Analysis is another introductory route through sequences, continuity, differentiation and integration. Alcock's Part 1 advice applies to all of these, and her Chapter 10 on completeness prepares for the axiomatic treatment of the real numbers that most course texts put early.

How to Use This Book

The book is written for students who have not yet started Analysis, for example in the summer before university or before upper-level courses in a US-style system. The author also hopes it will help students already in an Analysis course who are struggling, even those who start reading only before examinations. School calculus is the working background: the content chapters build on familiar ideas of sequences, series, differentiation and integration.

A practical plan:

• Start with Part 1. The introduction recommends that every reader begin there, even those with some university mathematics, because later chapters refer back to its advice. Chapters 2 and 3, on definitions, theorems and proofs, are the core.
• Read Part 2 in the order of your course. Use the overview at the start of each chapter to match it to the course: Chapters 5 and 6 for a sequences-and-series course, Chapters 7–9 for continuity, differentiability and integrability, and Chapter 10 wherever the real numbers are treated.
• Read actively. Try the self-explanation method from Chapter 3 on the proofs in Part 2. Test each definition on examples before reading on, as Chapter 2 recommends.
• Do not get stuck. Following the author's advice, mark a difficult section and move on; every chapter mixes harder and easier material.
• Use it alongside a textbook. Use it with the course text, not instead of it: once a chapter here makes sense, work the corresponding sections and exercises in Abbott or Bartle and Sherbert.