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Undergraduate Analysis

Aisling McCluskey and Brian McMaster
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Tom Schulte
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This introduction to analysis is suitable for students who have “a familiarity with elementary calculus.” It is assumed the reader has “knowledge of how to differentiate basic exponential and trigonometric functions,” etc. The focus is convergence and limits of sequences and series (more than half of the chapters), as well as differentiation (a couple chapters) and a smaller foray into integration. The Cauchy condition for convergence receives a short, dedicated chapter. Other topics include L’Hôpital, Lipschitz, a gamut of mean value theorems, and Taylor series. The authors’ main concerns come across as a firm, even rigorous, foundation in limits and convergence theory.

Called here the “epsilontics game”, the \(\varepsilon\)-\(\delta\)-definition of limit is grown from hints and exercises over several chapters into a formalization of the notion of limit. “Epsilon” itself is first introduced on page 13 in approximation and the concept is enlarged upon throughout in an approach that is considered and patient:

…mathematics is hard. By that we do not mean that it is intrinsically difficult: in this sense, “hard” is the opposite of “soft,” not the opposite of “easy.” Learning a piece of mathematics requires a precise understanding of the terms that it involves, of the arguments that it employees, and of the questions that it seeks to answer.

Ample space and explanations are given to the fundamental ideas of the topic at hand through a comprehensive range of examples and exercises. Some are worked in full detail, others are supported by sketch solutions and hints, while some are left wholly to the reader. (“Specimen solutions” are available to instructors by application to the publisher.) All are strategic cases selected appropriately and building one on the other.

…a lack of gradualness comes into play here, Most topics…. can be adequately explained to the beginner by working initially on simple special cases.

The use of informal, heuristic, even imprecise initial explorations of problems eases the reader into more rigorous approaches. Be it a “draft solution”, “partial draft solution”, or “rough work”, this is a measured introduction into detailed proof-building. Proofs support many key theorems, some with complete alternative approaches.

However, the arrangement of topics at first review brings back bad memories of passing out syllabi with a near derangement of chapters to concerned students. From “A Note to the Instructor”:

how this material is divided … across the semesters will vary from one institution to another.

Indeed. Limits of a sequence precedes a review of set theory and interval notation. Page 36 informs us “The exponential function … equals its own derivative”, while the derivative is defined on page 203. The introduction of the derivative is worth the wait, however. Building from the concept of the limit taken into the realm of functions starting 50 pages prior, the derivative is convincingly defined from obtaining the tangent through a limit approximation more artfully and convincingly than I often see.

While it is not given much space, a section introduces trigonometric functions through integration to find areas of the unit circle and is worth consideration as a classroom capsule itself. Such is the coherence and completeness of this logically self-contained introduction to analysis, that at the conclusion, regardless of the order taken, the payoff is

now having a logically watertight definition of integral …. We can at least provide reliable definitions of the so-called elementary functions such as \(\ln x\), \(e^x\) and \(\sin x\).

Indeed, the fusion of area under the curve, the natural log, and the divergence of the harmonic series in a way graspable to the undergraduate is a triumph compared to similar texts I have perused.

While I do have some doubts about recommending this as a sole textbook for a course, due to the organization of the material, this a suitable choice for the student seeking to make the transition to higher mathematics and aspiring to a graduate degree in a branch of pure mathematics as a reference text with value for several semesters.

Tom Schulte has taught undergraduates face to face for over eight years and is now transitioning to online classrooms.

1. Preliminaries
2. Limit of a sequence, an idea, a definition, a tool
3. Interlude: different kinds of numbers
4. Up and down - increasing and decreasing sequences
5. Sampling a sequence - subsequences
6. Special (or specially awkward) examples
7. Endless sums - a first look at series
8. Continuous functions - the domain thinks that the graph is unbroken
9. Limit of a function
10. Epsilontics and functions
11. Infinity and function limits
12. Differentiation - the slope of the graph
13. The Cauchy condition - sequences whose terms pack tightly together
14. More about series
15. Uniform continuity - continuity's global cousin
16. Differentiation - mean value theorems, power series
17. Riemann integration - area under a graph
18. The elementary functions revisited