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Closer and Closer: Introducing Real Analysis

Carol S. Schumacher
Jones and Bartlett
Publication Date: 
Number of Pages: 
[Reviewed by
John D. Cook
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Closer and Closer is a new textbook for undergraduate real analysis. There is near-universal agreement on what an undergraduate class in real analysis should cover, and the table of contents of this book corresponds to that consensus. Closer and Closer stands out not for its choice of topics but for how those topics are presented.

Closer and Closer is divided into two parts: “central ideas” and “excursions.” The central ideas are the main theory of the real analysis, while the excursions are common examples and applications. A glance at the table of contents would suggest that the two parts are to be covered sequentially. That is not the intention. The excursions are meant to be covered thoughout the course of the main development. The chapters include pointers directing students to specific excursions once they have sufficient background.

Why separate the applications from the main development? This doesn’t seem necessary, but there are perhaps a few advantages. First, it may help students distinguish the basic theory from specific examples. Second, it could help instructors decide what material to cut if a course is running out of time. On the other hand, most the material in the excursions is essential. Labeling these topics as “excursions” might imply that they are less important than they actually are.

The strength of Closer and Closer is its exposition. Schumacher does a fine job of proving theorems rigorously, but also provides intuitive explanations and motivation. While such exposition is appreciated in any mathematics book, it is especially important in real analysis. A course in real analysis is of course supposed to present the basic results of real analysis. But it also does much more. It teaches students to take intuitive ideas and express them as rigorous mathematical statements. Other courses do the same, but in general it is more difficult to formalize continuous mathematics than discrete mathematics.

The history of mathematics bears this out. The material in a rigorous real analysis course represents the conclusion of decades of research and heated debate on the foundations of continuous mathematics. It is especially important in real analysis to explain the correspondence between formal mathematics and intuitive notions.

Closer and Closer contains in written form much of the dialog given in lecture by a good instructor but not often committed to paper. Having such good explanations in the textbook may allow an instructor to devote a little more class time to discussion and problem solving.

John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs daily at The Endeavour.


Part I     Central Ideas

Preliminary Remarks
0.   Basic Building Blocks
1.   The Real Numbers  
2.   Measuring Distances
3.   Sets and Limits
4.   Continuity
5.   Real-Valued Functions
6.   Completeness
7.   Compactness
8.   Connectedness
9.   Differentiation of Functions of One Real Variable
10.  Iteration and the Contraction Mapping Theorem
11.  The Riemann Integral
12.  Sequences of Functions
13.  Differentiating f: Rn - R

Part II  Excursions

1.   Truth and Provability
2.   Number Properties
3.   Exponents
4.   Sequences in R and R
5.   Limits of Functions from R to R
6.   Doubly Indexed Sequences
7.   Subsequences and Convergence
8.   Series of Real Numbers
9.   Probing the Definition of the Riemann Integral
10.  Power Series
11.  Everywhere Continuous, Nowhere Differentiable
12.  Newton's Method
13.  The Implicit Function Theorem
14.  Spaces of Continuous Functions
15.  Solutions to Differential Equations