This aptly named beginning textbook in Real Analysis gets to many important ideas in analysis without attempting to address them all, completely skipping topology for example. The choice of what is important is motivated by the mindset of an applied analyst, with nicely chosen optional applied topics.
The text focuses on one real variable and is accessible to the typical math major taking a first course in analysis, so that is at the level of texts by Bartle and Sherbert or Kenneth A. Ross. It builds a solid foundation in sequences but avoids topological perspectives, in order to get on to continuity and a brief treatment of Riemann integration that precedes differentiation. A typical course should make it through to Chapter 5 on Sequences of Functions that introduces the supremum norm and the space C[0,1] and even discusses the problem of completeness using p-norms. My course did not cover the end of this Chapter with sections on Metric Spaces, the Contraction Mapping Principle and Normed Linear Spaces, though the inclusion of those topics shows the mindset of this approach. In fact, the first definition of completeness is in terms of Cauchy sequences, instead of sup and inf, a broad perspective on analysis that I like, although a different ordering can allow more efficient proofs. Bisection is used to prove completeness in terms of monotone sequences, supremum, and Bolzano-Weierstrass. Discussion of lim-sup and series is delayed until Chapter 6 on Series of Functions, again employing a kind of just-in-time approach that uses series to get to the broader picture of functions.
Optional sections on applications motivate and supplement the text. The topics are interesting: Markov Chains and the Quadratic Map in the beginning chapter on Sequences, Numerical Methods using calculus, and Integral Equations within Sequences of Functions. The latter feeds into the first of the optional chapters on Differential Equations, Complex Analysis, Fourier Series, and Probability Theory. The small sections dedicated to such topics are sometimes not very satisfying, since they often center on one very technical proof that doesn’t give a good sense of the topic. Nevertheless, an instructor can pick one (time rarely allows for much more) and introduce or expand that topic of personal interest. I use the section on the Quadratic Map to show students that sequences do not have to be tame but that simple iteration leads to the fascinating bifurcation to chaos diagram (this text mentions the idea but not the diagram). Instead of the section’s technical proof, I return after the Mean Value Theorem to prove convergence to a fixed point if the derivative is less than one. Section 3.4 on Numerical Methods is misplaced, since it is an application of Taylor’s Theorem, treated in Section 4.3.
The attempt to provide only what is needed usually works, with one exception in limits of functions. After continuity is done well using only sequences and ε–δ, differentiation uses limh→0. This concept is only covered in half a page that defines limx→c . After convincing students that everything must be justified by a definition or theorem, this Chapter acts as if you can do anything that seems reasonable with limh→0 notation. At first, I tried to use only sequence theorems with hn→0 as n→∞, but students followed the text. On my second offering, I handed out a supplement of how all of the limit theorems for sequences give limit theorems for functions. I wish the text would have chosen one of these options.
Accessibility and efficiency is favored over the most general statements of theorems. In the Riemann Integral section, theorems assume that functions are continuous. This is all right with me, but the inequalities are actually easier to prove using just the assumption that the function is Riemann integrable.
The book is well written with good prose that really explains ideas and the bigger picture. There are some quirks (such as every ε inequality uses ≤, instead of <) and a small number of typos (such as missing absolute values around h in 0 < h < δ derivative characterization). I was pleased with the nice variety of exercises and projects. Overall, I’ve found that the text worked well for teaching a range of student abilities. The best students were prepared to go on to a second course using the text by Saxe on Beginning Functional Analysis.
Richard Neidinger is professor of mathematics at Davidson College in North Carolina. Trained in functional analysis, his interests include numerical analysis, dynamical systems, and undergraduate mathematics education. Some of his favorite topics are automatic differentiation, discrete chaos, and fractals. Email: email@example.com.
The Riemann Integral
Sequences of Functions
Series of Functions