Many undergraduate students who are taking real analysis (or advanced calculus) for the first time struggle with two major obstacles: understanding numerous new abstract concepts, and dealing with these concepts at the level of rigor necessary for their mathematical arguments. For example, the \( \epsilon \)-\( \delta \) definition of continuity contains multiple nested quantifiers:

Given a set of real numbers \( D \), a function \( f : D \rightarrow R \) is continuous at \( c \in D \) if for all \( \epsilon > 0 \), there exists \( \delta > 0 \) such that for all \( x \in D \), if \( |x − c| < \delta \), then \( | f(x) - f(c) | < \epsilon \).

When students are asked to prove that a function is continuous for the first time, many will look at this definition and have no idea where to get started. To aid students in overcoming these obstacles, the author presents “Proof Strategies” throughout the book. For example, after presenting the \( \epsilon \)-\( \delta \) definition of continuity, the author decomposes the definition into the following “Proof Strategy”:

Let \( f : D \rightarrow R \) be a function and let \( c \in D \). To prove that \( f \) is continuous at \( c \), use the proof diagram:

Let \( \epsilon > 0 \) be an arbitrary real number.

Let \( \delta = \) (the positive value you found).

Let \( x \in D \) be arbitrary.

Assume \( |x − c| < \delta \).

Prove \( |f (x) − f (c)| < \epsilon \).

The author then demonstrates this strategy by formulating an argument that the function \( f : R → R \) given by \( f(x) = 3x^{2} + 5 \) is continuous. Within an environment the author calls a “Proof Analysis”, they explain how to choose an appropriate value of \( \delta \) for the proof strategy above. After this exposition, the author presents a cogent, formal proof that f is continuous. Similar “Proof Analyses” precede the proofs of major theorems throughout Chapters 2 - 4. This way of presenting the material is helpful for students, but it is also quite useful for instructors who are new to teaching real analysis.

The exercises at the end of each section include suggestions for which proof strategies from the text to employ. Furthermore, the author also includes “Exercise Notes” which provide hints for the trickier exercises.

This textbook is intended for undergraduate students who have completed a standard calculus course sequence that covers differentiation and integration and a course that introduces the basics of proof-writing. For students who have a limited proof-writing background, the author includes an abbreviated discussion of proofs, sets, functions, and induction in Chapter 1.

Because the author focuses on proof-writing skills, the amount of illustrations and figures in the book is rather modest. Consequently, the exposition would be best paired with drawings made by the instructor or animations from online resources.

In summary, this book is a good resource for student’s who are taking a first course in real analysis and who have a limited background in proof-writing.

Lawrence Mouillé is currently a G.C. Evans Instructor in the Department of Mathematics at Rice University. He received his Ph.D. in mathematics from the University of California, Riverside in 2020.