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Closer and Closer: Introducing Real Analysis

Author(s): Carol S. Schumacher, Kenyon College
  • ISBN-13: 9780763735937
  • ISBN-10:0763735930
  • Paperback    438 pages      © 2008
Price: $149.95 US List
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Closer and Closer is the ideal first introduction to real analysis for upper-level undergraduate mathematics majors. The text takes students on a guided journey through the often challenging world of analysis, providing them with the tools to solve rigorous problems with ease. The author achieves this with a student-friendly writing style, an active learning approach, and rich examples and problem sets, along with a unique two-part format. Core Chapters open the text and introduce the most important tools used in analysis. The Excursions then round out and complement Core chapters, allowing students to explore new problems on their own. This two part approach provides a flexible, interactive introduction to relevant concepts and allows students to truly understand and retain key material presented throughout the text. Closer and Closer offers an unparalleled introduction to the foundations of this important area of mathematics.


Features & Benefits

Boxed asides provide tips on proof techniques, comments about the significance of theorems, and remarks about notational conventions.

The text bridges the gap between intuition and abstraction by including many easy exercises and examples that connect abstract concepts to the student’s own understanding of more concrete ideas.

Explanatory notes accompany some problems and explain the context of the problem or invite the student to reflect upon the implications of the result.

Flexible two-part structure is ideal for diverse groups of students as well as instructors having various and differing requirements.


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 - Rm

Part II  Excursions

1.   Truth and Provability
2.   Number Properties
3.   Exponents
4.   Sequences in R and Rn
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 


Carol S. Schumacher-Kenyon College

Carol S. Schumacher got her undergraduate degree at Hendrix College in 1982 and her PhD in Mathematics from the University of Texas at Austin in 1989.  She joined the faculty of Kenyon College in 1988 and is currently Professor of Mathematics there.  Professor Schumacher is the recipient of Kenyon's Trustee Teaching Award and the author of Chapter Zero - Fundamental Notions of Abstract Mathematics, 2e (A-W, 2001).  She was born in La Paz, Bolivia and Lives in Gambier , Ohio with her husband and two daughters.

  • The main objective of letting the students do [!] the proofs of  the main results themselves is remarkably well achieved. The key  is clearly to break these problems into reasonable chunks - the  author’s experience of what that appropriate size is has been  invaluable to me.


    Matthias Kawski   

    Dept. of Mathematics and Statistics        
    Arizona State University


The following instructor resources are available to qualified instructors for download

ISBN-13: 9780763735937

Answers to In-Text Questions
Instructor Manual