Introduction to real analysis. (English) Zbl 0856.26001

Englewood Cliffs, NJ: Prentice Hall. xiii, 368 p. (1996).
This is an introductory-level real analysis text for students who have completed a calculus course. (In the U. S. the latter is typically manipulative and proof-free.) It is meant to be the student’s entrée into mathematics as a logical enterprise, so the construction, analysis and dissection of proofs is a continuing and central theme. Naturally, considerable effort is devoted to securing the real numbers; they are not simply axiomatically summoned. The idea of a complete, archimedean-ordered field evolves, and all the major highways and feeder roads are carefully constructed. The myriad equivalent manifestations of completeness (compactness, connectedness,...) are collected in what the author calls The Big Theorem, its order of battle in a useful diagram called The Big Picture. This enterprise is motivated (on p. 1) by The Big Question – how are the real numbers different from the rational numbers? These highway signs reappear regularly to keep the reader on course. The existence question itself is settled in a brief final chapter via cuts \((\mathbb{Q}\) considered as given), but the (better) alternative via equivalence classes of Cauchy sequences in \(\mathbb{Q}\) is dicussed in an exercise, as are the constructions of \(\mathbb{Z}\) from \(\mathbb{N}\) and of \(\mathbb{Q}\) from \(\mathbb{Z}\). (But Peano and the genesis of \(\mathbb{N}\) are not mentioned.)
Chapter 1 is on logic, connectives, quantifiers, sentence-writing, set-theoretic operations and proofs. These’s lots of good advice and examples here. (But why is the possibility of a third truth-value, “matter-of-opinion”, illustrated with a question: “It’s a nice day, isn’t it?”?) There is valuable discussion on placement of quantifiers; but this important theme could have been profitably reprised at a higher level following T. Whaley and J. Williford [Am. Math. Mon. 87, 745-788 (1980; Zbl 0475.26004)]. This chapter is a miniature of D. Solow’s successful book “How to read and do proofs: an introduction to mathematical thought processes” (2nd ed. 1990; Zbl 0711.00001)], which it references. Chapter 2 deals with equivalence relations and cardinality \((\mathbb{N}\) and \(\mathbb{Q}\) considered given). One of its exercises is an interesting variant of Richard’s Paradox. Chapter 3 is on algebraic structure of fields \((\mathbb{R}\) used freely as an example). Chapter 4: orderings, proof by induction, ordered fields, intervals and neighborhoods. As The Big Picture is filled out in Part II (Chapters 5-12), topological issues, convergent sequences, and continuous functions are introduced and analysis begun. Actually the general concept of a topology on a set is defined and this enables the author to later streamline the treatment of continuity on subsets of \(\mathbb{R}\) that are not open, by employing the relative topology (for which, however, his notation is a bit unorthodox). Construction and properties of the Cantar set appear as a multi-part exercise, as do the one-point and two-point compactifications of \(\mathbb{R}\).
Part III is called “Topics in Calculus” and its six chapters deal with numerical series, uniform continuity, power series and the topology of \(C[0,1]\), differentiation and integration (Riemann integration based on refinement, with no mention made of the alternative mesh theory). As in earlier chapters, some facts known from calculus but not yet rigorously grounded enter into exercises and examples. Other topics, some treated as exercises with hints, include Raabe’s test, Riemann’s rearrangement theorem, Mertens’ theorem, the Gregory-Leibniz series, infinite products and bilateral series, the Arzelà-Ascoli theorem, and functions of bounded variation. The central problem of analysis, interchanging two limit processes, is the theme of Chapter 18; included are double series, differentiation and integration of sequential limits and of integrals involving a parameter. Here we also find a useful, not so well-known theorem of E. H. Moore. Its intricate proof would have been the quintessential vehicle for the author’s considerable pedagogical skills, but it is withheld. Part IV is selected short subjects: discontinuities of monotone functions, the Cantor-Lebesgue singular function and mention of Baire category and measure-zero sets. Baire’s theorem is not proved. Lebesgue’s criterion for Riemann integrability is stated (but is missing the essential boundedness hypothesis). Other unusual features of the book: Numerous proofs are built from both ends, like the chunnel, with questions asked and methods evaluated and rejected en route to closure. The author is admirably punctilious about quantifiers, and avoids the unfortunate adjective “non-decreasing” (as Dieudonné pointed out, sin\(x\) is nondecreasing – it’s not a decreasing function). Jensen’s integral inequality, Newton’s method, the Antipodensatz for the circle, continued fractions and the contraction mapping principle (with an application to the iterates of the cosine function) appear as exercises. So, without warning labels, do the tower of powers problem \((x_{n + 1} : = x^{x_n})\) and the equality \(\varlimsup \cos (n) = 1\). Being hintless, these seem beyond the anticipated reader of this book. (One needs to know that infinite subsemigroups of the circle are dense and that \(\{e^{in} : n \in \mathbb{N}\}\) is such a semigroup, due to the fact – not mentioned in the text – that \(\pi\) is irrational.)
Altogether this is an excellent, “user-friendly” book, which the reviewer would happily prescribe as a course text. The writing style is unhurried, lively, informal (as Landau put it, the reader is “geduzt”), at times colloquial, with some good word plays (e.g., “a new slant on the derivative”). This said, balance requires the reviewer to offer some (mild) criticisms. There are occasional linguistic lapses: “Alternate” is used where “alternative” is meant. (“Real and complex analysis are offered in alternate semesters, but alternative courses are available.”) A cautious and uncertain “conditional imperative” [“If \(P\), then prove that...”] sometimes displaces the unqualified injunction, more proper in mathematics, to prove a conditional [“Prove that, if \(P\), then...”]. Due either to oversight or misguided “political correctness” (an American dementia), the author favors plural pronouns with singular subjects like “the student”, perhaps to the confusion of international readers. Theorem 19.6 affirms that “Any (sic) function can have only countably many jump discontinuities,” and its proof contains the expression “\(|f(x) - \lim_{x \to a^+} f(x) |< \varepsilon/2\) when \(x \in (a,a + \delta)\)”. Although formally correct, the dual role of \(x\) here will needlessly waylay the reader’s train of thought. There is no symbol index (the reader is challenged to find the author’s somewhat unconventional definition of the symbol \(\hookrightarrow)\), and page references in the general index are red-shifted – by an unknown but strongly monotone function – rendering it almost useless. (The reviewer has learned that this problem was generated by the editors and will be remedied in a second printing.) The greatest resource of mathematical pedagogy, the American Mathematical Monthly, is cited only twice, the Mathematics Magazine not at all. Relevant articles there would greatly help the reader with the aforementioned tower of powers, as well as with the author’s exercise on the Kuratowski closure-and-complementation problem [see J. Berman and S. L. Jordan, Am. Math. Mon. 82, 841-842 (1975; Zbl 0312.54003) and J. H. Five, Math. Mag. 64, No. 3, 180-182 (1990; Zbl 0735.54001)]. The author offers some good examples of erroneous “proofs,” but many more could profitably have been culled from the “Flim-Flam” section of The College Mathematics Journal. B. R. Gelbaum and J. M. H. Olmsted’s indispensable classic “Counterexamples in analysis” (1964; Zbl 0121.28902)] that every beginning student of analysis needs to make the acquaintance of, is not mentioned by the author. In discussing \(x^n\) \((0 \leq x \leq 1)\) and uniform convergence, the author rightly reminds the reader that despite the prevalent phrase “uniformly Cauchy”, the name of this “brilliant” mathematician is not an adjective. The irony of Cauchy’s having missed the uniformity concept and having erroneously claimed that pointwise convergence preserves continuity might also have been noted here. And the “bland traditional terminology Weierstrass \(M\)-test” could have been demystified by reference to “Majorant”. The transcendental functions are not developed (a significant lacuna, the review feels), nor is l’Hospital’s rule mentioned.


26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions