There are many elementary texts on real analysis that describe themselves as “introductions” to real analysis but this text goes a step further and describes itself as an “invitation”. This word caught my interest because I think that an invitation promises to be more than a mere introduction. It suggests that the book will be charming, gentle, and reassuring and that it will be written in a way that whets the appetite of the reader. However, I am sorry to say that this text by Cesar Silva does not meet that standard. In my opinion, a student is likely to find this book to be difficult, scary, and nearly unreadable. It is anything but inviting.

My first concern about this book is about the flow of its language. Not only is it labored and unfriendly, but it frequently contains meaningless phrases that ignore the basics of English grammar. This is unfortunate because I think that an important role of a beginning course in real analysis is to teach precise mathematical writing and careful justification of mathematical statements. I list just two of what I consider to be unacceptable instances of mathematical writing in this text. In Section 0.5.6 we are told that “the integers consist of the natural numbers, the number 0, and the negative natural numbers”. The “integers consist”? I think the author may have meant to say that the set \( \mathbb{Z} \) of integers contains the numbers he mentions. Furthermore, what on earth is a negative natural number? On page 47 we are told that “the real numbers are defined to be an ordered field that is complete under that order”. The English grammar of this definition does not make sense. This item is just one of many examples in which real numbers are confused with the set of real numbers, integers are confused with the set of integers, and so on. Later on, the book refers to the “topology of the real numbers” rather than to the topology of the number line. The book refers to "construction" of the integers or of the rationals or of the reals, rather than to "the construction" of the systems of integers, rationals, or reals.

I am concerned about the variety of advanced topics in this book that seem to be presented in the wrong context, or without context, or in the wrong chapter, and that often go nowhere. I found myself wondering why these topics were included at all. Time and time again, I battled to make sense of what I was reading, only to discover that the key definitions were still to be presented in a later part of the book. Instead of inflicting these advanced topics on a struggling reader, the book should be presenting the basic material much more carefully and gently to make it friendly and inviting. I list two examples of such advanced topics:

In Section 0.5.8, after a discussion of factorization properties in the system of natural 1numbers, the book suddenly presents the Cantor-Schröder-Bernstein theorem. The proof given may have historical interest but it is clumsy and labored and, unfortunately, it presupposes the system of natural numbers. Thus the presentation of this theorem is not forward compatible with a later course in set theory. Any student who wishes to move forward in the future will have to study this theorem again from scratch to allow the presentation of the set theory to be used as a basis for constructions of the number systems. Although the book makes a couple of minor applications of the equivalence theorem in Example 0.5.19, it misses many golden opportunities to simplify material and obtain interesting facts by using the equivalence theorem. A case in point is Lemma 1.3.3 which has a clumsy proof that could have been simplified by using the equivalence theorem. The book omits many interesting and simple applications of the equivalence theorem leading me to ask why the equivalence theorem was presented at all.

The presentation of the Cantor set appears in Chapter 3 together with a difficult and unmotivated section on sets of measure zero. These topics also make use of infinite series that are introduced in Chapter 7 and so, if these topics have to be presented at all, they should appear much later. I would have thought that a purpose of presenting the Cantor set and the measure zero concept might be to show their roles in Riemann integration theory but the book doesn’t do that. So why were these topics presented at all?

Several basic topics are incomplete. For example, the presentation of continuous functions on intervals in Section 4.2 omits some key theorems that are standard fare in a first course in real analysis. The book omits the fact that a monotone function on an interval is continuous if and only if its range is an interval, the fact that an injective continuous function on an interval must be strictly monotone, and the fact that an injective continuous function on an interval has a continuous inverse function.

Finally, I was disappointed to see that this text is one of those many books that present the “nested intervals theorem”. It appears in Section 2.2 and tells us that the intersection of a contracting sequence of nonempty closed bounded intervals must be nonempty. In fact, intervals are irrelevant in this theorem. In the system \( \mathbb{R} \), it is no harder to prove the useful Cantor intersection theorem that says that the intersection of a contracting sequence of nonempty closed bounded sets must be nonempty. By confining its attention to intervals, the book loses the opportunity to present some nice applications of the Cantor intersection theorem.

To sum up, I’ll say that the writing of this book should have been more precise, more polished, and more friendly. The book should present a clear idea of what it wants to teach and it needs to present those topics more completely. Instead of giving half hearted, unmotivated, and incomplete discussions of advanced topics, it should stick to what we can actually expect the students to study and understand.

Jonathan Lewin is currently Professor of Mathematics at Kennesaw State University where he has been for the past 37 years. He did his undergraduate study at the University of the Witwatersrand in South Africa and then obtained his PhD at the University of Wisconsin in 1970. He has held faculty positions in South Africa and Israel, and in the USA. He is the author of several texts including a beginning real analysis text that is published by Cambridge University Press. He is also the author of five graduate level textbooks that are supplied as public domain works for on-screen reading.