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The Real Numbers and Real Analysis

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The most distinctive characteristic of this text on real analysis is its three-in-one feature. It was designed specifically for three distinct groups of students. The first audience consists of mathematics majors taking an ordinary introduction to real analysis. The second is prospective high school mathematics teachers getting an introduction to real analysis. The third group is prospective high school teachers taking a second real analysis course, presumably as part of a Master of Arts in Teaching (M.A.T.) degree. The book was motivated by a need for a textbook for the M.A.T. students, but is intended to have enough flexibility to serve the other groups as well. The author provides suggested paths through his book for each potential audience.

The topics treated are more or less standard. Beginning with the construction of real numbers and their properties, the author proceeds through limits and continuity, derivatives and integrals. Transcendental functions are treated in some detail later in a separate chapter. Sequences and series — first of numbers, then of functions — are presented at the end. The treatment of the real numbers in the first two chapters (more than a hundred pages in total) is a good deal more extensive than in comparable texts. It includes, for example, the Peano postulates, axiomatic treatment of the integers, rationals, and real numbers, and construction of the reals via Dedekind cuts.

The approach is detailed and rigorous, and the level of sophistication is in the middle of the spectrum. The author is a believer in “slow and steady”, so the proofs he provides are usually written out in full detail. He also prefers to minimize technicalities, so he omits things like limits inferior and superior, and avoids proofs that are too slick. Because sequences and series are not treated until the end, some theorems about continuity or derivatives, for example, have somewhat more tedious proofs than they might otherwise.

This is a talky book. The author includes historical remarks in every chapter and “Reflections” in every section. The purpose of the reflections is to try to give the student a broader perspective on what he or she is learning. In general, these additions are admirable and desirable, but combined with a wordier style throughout, the text occasionally has a flavor of “too much-ness”. While one might readily choose this over a terse style like that of Rudin’s, there is clearly a tradeoff. Students, in my experience, do not readily read mathematics textbooks, so more is perhaps not better.

I have learned to be wary of historical background material in mathematics texts. I am by no means an expert, but I fear that too many authors uncritically pass on historical errors and misconceptions. While I have no special concerns about the historical notes in this text, there does seem to be a tendency in this book to portray the history of calculus as an onward and upward path without the missteps and dead ends we all know are a real part of mathematical development.

As I find with many other introductory analysis texts, I am troubled again here by the tendency to treat the course as a means of ratifying the theorems of calculus. As Thomas Kőrner says in A Companion to Analysis, “It is surprising how many people think that analysis consists in the difficult proofs of obvious theorems.” I feel very strongly that students need to understand why they are pushed to this level of rigor, what can go wrong, and why a little doubt is a good thing.

In spite of my quibbles, this is a strong text, especially for students who need more guidance and support. The book gives an instructor plenty of options for planning a course.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Date Received: 
Wednesday, June 8, 2011
Include In BLL Rating: 
Ethan D. Bloch
Publication Date: 
William J. Satzer
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Tuesday, September 27, 2011