Undergraduate Analysis

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.