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Lectures on Real Analysis

Finnur Lárusson
Cambridge University Press
Publication Date: 
Number of Pages: 
Australian Mathematical Society Lecture Series 21
[Reviewed by
Allen Stenger
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This is a clear, concise, compact, and dry first course in real analysis, aimed at proving all the facts used in single variable calculus, sequences and series. It is intended as a second course in mathematics, after calculus, but for American audiences it seems to assume too much mathematical maturity for this to succeed.

The book is well-equipped with exercises. A few are scattered in the body, where they illustrate some point that has just been discussed, but most are at the ends of chapters and ask for proofs of calculus facts that are well-known to mathematicians but are very challenging for students at this level.

In addition to the traditional epsilon-delta approach, it uses a more conceptual look at limits through neighborhoods, and overall the book has a slightly topological or metric-space flavor. It discusses completeness of the reals in detail, although adopting it as an axiom rather than constructing the reals from the rationals. There is a development of the trigonometric functions after defining them by power series. There is some material at the end on complete metric spaces, contraction mappings, and Picard’s theorem on the existence of solutions to first-order differential equations.

The present book is what you would get if you took a rigorous calculus book and removed all the pictures, examples, applications, and motivation. It is thus comparable to Hardy’s A Course of Pure Mathematics and Landau’s Differential and Integral Calculus, although those books teach the mechanics of calculus as well as its theory. Another somewhat comparable book, probably more agreeable to modern audiences than those two classics, is Boas’s A Primer of Real Functions. Boas’s book is ordered differently, because it is looking forward to analysis and not backward to calculus, but it covers the same topics and more. Boas would be a better book for most introductory real analysis courses, because it is livelier, more scholarly, gives a better flavor of real analysis, and is only slightly more expensive.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

To the student
1. Numbers, sets, and functions
2. The real numbers
3. Sequences
4. Open, closed, and compact sets
5. Continuity
6. Differentiation
7. Integration
8. Sequences and series of functions
9. Metric spaces
10. The contraction principle