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A Basic Course in Real Analysis

Ajit Kumar and S. Kumaresan
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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This is a textbook for a course in single-variable real analysis at the junior/senior undergraduate level. The syllabus for such a course has by now become something of a sacred cow, and is tracked faithfully by this book’s contents, which, in order, cover: properties of the real numbers, sequences, continuity, differentiability, infinite series, integration (here, Riemann only, neither Riemann-Stieltjes nor Lebesgue), and sequences and series of functions. The book ends with a statement and proof of the Weierstrass Approximation Theorem (using Bernstein polynomials).

As the title of the book indicates, the authors take pains to keep things basic and avoid some of the more difficult and subtle ideas that might have been included. For example, although basic properties of the real numbers are specified (essentially, that they form an ordered field satisfying the least upper bound property), no attempt is made to actually construct the real numbers by Dedekind cuts or equivalence classes of Cauchy sequences. In addition, one topic (the lim inf and lim sup of a bounded sequence) that is often done in the main part of a text is here relegated to an Appendix. Although opinions may differ, I think both of these are wise choices on the part of the authors.

I am much less in favor, however, of the authors’ decision to omit certain other topics from the text. Metric spaces, for example, are not mentioned; I think an introductory course in real analysis should contain at least some exposure to this concept. If metric spaces in that level of generality are not introduced, then I would think some discussion of the basic topology of the real numbers should be, but that is not done in this book to the extent that I would like. Although the authors state and prove the Intermediate Value Theorem, I didn’t see any discussion of how the result is not true if the domain is not an interval, or why intervals are different from other kinds of subsets of the real numbers; the reader will need to look elsewhere for a discussion of the concept of connectedness. For the Extreme Value Theorem, however, the authors do point out examples showing the necessity of the assumption that the domain is a closed bounded interval, and briefly discuss compactness (defined here not in terms of open covers but convergent subsequences); however, the discussion is not extensive, and does not, for example, include the Heine-Borel theorem. The Baire Category theorem also does not appear, and neither do familiar constructs like the Cantor set. The extent to which an instructor might find this problematic is, of course, a matter of individual taste, but I think there has been a little too much omitted here.

Back in February 2013, when I reviewed Taylor’s Foundations of Analysis for MAA Reviews, I wrote that “given that there are lots of books in this area of mathematics and that most of them cover pretty much the same material, it would seem that any new real analysis book should have some sort of distinctive feature to it, in order to set it apart from the pack.” The intended “hook” or distinctive feature of this text is an especially reader-friendly style of writing, intended to allow a prospective student to read the book without professorial assistance.

Several techniques are used to achieve this goal. One such technique, referred to above, is the omission of some of the more sophisticated topics that one might expect to find in a book like this. In addition, the authors write in a conversational and accessible tone, they provide lots of examples worked out in detail. Geometric insight is stressed throughout, and, in what I thought was a very nice touch, many proofs are prefaced by an informal discussion entitled “strategy” that provides a motivating overview of the proof that is to follow. There are also quite a few exercises, spanning a wide range of difficulty; hints (but not complete solutions) are provided for a selection of them at the end of the book in an Appendix section.

Another Appendix section talks about basic quantifier logic, with an emphasis on negating quantified statements. This is valuable, but I wish this section had gone further and also discussed the difference between a statement and its converse, as well as basic proof techniques such as proof by contradiction and proof by contrapositive. More and more, it seems, classes in real analysis are being populated by students who have just not fully assimilated these ideas and could use a section on them to turn to for reference.

As a result of the techniques described above, the authors have produced a book that is, indeed, genuinely reader-friendly. On only a few occasions, reading it, did I think there were easier ways to present the material. One example is the “sandwich lemma” (often referred to as the “squeeze lemma”, asserting that if the terms of one sequence lie between the terms of two others, both of which converge to the same number L, then the original sequence also converges to L). I thought the proof given in the text could have been simplified by reducing to the case when the “leftmost” sequence is identically zero. I also wish more books would employ a technique for doing limit arguments that I first saw explicitly described in Mattuck’s Introduction to Analysis; he refers to it as the “K-\(\varepsilon\) principle” and describes it thusly: “let’s once and for all agree that if you come out in the end with 2\(\varepsilon\) or 22\(\varepsilon\)that’s just as good as coming out with \(\varepsilon\)If \(\varepsilon\) is an arbitrary small number, then so is 22\(\varepsilon\). Therefore, if you can prove something is less than 22\(\varepsilon\), you have shown it can be made as small as desired.” This principle, though mathematically obvious, saves a lot of time and effort in doing limit theorems, and, I think, also helps clarify for the student what “\(\varepsilon\)-arguments” really mean.

These kinds of examples, where I thought something could be made simpler, are rare. By and large this book describes the basic results of analysis in an extremely clear, straightforward, and well-motivated way.

On one or two occasions, there were definitions that struck me as somewhat non-standard. For example, the authors define an interval to be any set of real numbers that contains every number between any two of its elements. They then list the standard types of intervals (bounded, unbounded, open, closed, half-open, etc.), and point out that they satisfy the defining condition. The converse is mentioned but not proved.

I’ll close by pointing out one other concern that I have, one that I also have with a number of other analysis books. Since many of the results that make up an analysis course are already more or less familiar to the student from his or her calculus days, albeit at a much lower level of rigor, it seems to me that the beginning of an analysis course should address the question of why we even need to make elementary calculus more precise. Most freshman calculus courses offer up an impressive array of applications of calculus to physics, biology or economics; why not just stick with that? What do we gain, other than confusion, by adding Greek letters like \(\varepsilon\) and \(\delta\) to the mix?

Several books, though not as many as I would like, do make an effort to address this question. Abbott’s Understanding Analysis, for example (which to my mind represents the gold standard among current analysis textbooks) begins each chapter with a section entitled “Discussion” that poses a problem or introduces an issue that motivates the rest of that chapter; for example, the first chapter, on the real numbers, begins by pointing out the “holes” in the rational numbers and the need to fill them in, and the next chapter on sequences and series starts by talking about the issues encountered when we try and rearrange the terms of a series. Körner’s A Companion to Analysis opens with a short but illuminating section titled “Why do we bother?” in which he points out how some of the familiar theorems of calculus fail over the rational numbers, thereby suggesting that we need to look more closely at the real numbers to see what they have that makes calculus work that the rationals don’t. And Bressoud’s A Radical Approach to Real Analysis, a text that is considerably more sophisticated than this one, begins with a good discussion of how Fourier’s work in the early 1800s forced mathematicians to re-evaluate their views on matters such as the definition of a function, and continuity. So, although the book under review does a fine job of motivating and explaining the details of individual proofs, it could perhaps have done more to motivate the need for all this stuff in the first place.

To summarize and conclude: if you’re looking for a text on the easy end of the spectrum for a course in real analysis, then this book is certainly worth a serious look, although some supplementation on the instructor’s part may, depending on individual preference, be necessary.

Mark Hunacek ( teaches mathematics at Iowa State University. 

Real Number System
Algebra of the Real Number System
Upper and Lower Bounds
LUB Property and Its Applications
Absolute Value and Triangle Inequality

Sequences and Their Convergence
Sequences and Their Convergence
Cauchy Sequences
Monotone Sequences
Sandwich Lemma
Some Important Limits
Sequences Diverging to
Sequences Defined Recursively

Continuous Functions
Definition of Continuity
Intermediate Value Theorem .
Extreme Value Theorem
Monotone Functions
Uniform Continuity
Continuous Extensions

Differentiability of Functions
Mean Value Theorems
L'Hospital's Rules
Higher-order Derivatives
Taylor's Theorem
Convex Functions
Cauchy's Form of the Remainder

Infinite Series
Convergence of an Infinite Series
Abel's Summation by Parts
Rearrangements of an Infinite Series
Cauchy Product of Two Infinite Series

Riemann Integration
Darboux Integrability
Properties of the Integral
Fundamental Theorems of Calculus
Mean Value Theorems for Integrals
Integral Form of the Remainder
Riemann's Original Definition
Sum of an Infinite Series as an Integral
Logarithmic and Exponential Functions
Improper Riemann Integrals

Sequences and Series of Functions
Pointwise Convergence
Uniform Convergence
Consequences of Uniform Convergence
Series of Functions
Power Series
Taylor Series of a Smooth Function
Binomial Series
Weierstrass Approximation Theorem

A Quantifiers
B Limit Inferior and Limit Superior
C Topics for Student Seminars
D Hints for Selected Exercises