A Problem Book in Real Analysis

1. Front Matter

   Pages i-vii

   PDF

2. Elementary Logic and Set Theory

   • Asuman G. Aksoy, Mohamed A. Khamsi

   Pages 1-19

3. Real Numbers

   • Asuman G. Aksoy, Mohamed A. Khamsi

   Pages 21-39

4. Sequences

   • Asuman G. Aksoy, Mohamed A. Khamsi

   Pages 41-62

5. Limits of Functions

   • Asuman G. Aksoy, Mohamed A. Khamsi

   Pages 63-76

6. Continuity

   • Asuman G. Aksoy, Mohamed A. Khamsi

   Pages 77-96

7. Differentiability

   • Asuman G. Aksoy, Mohamed A. Khamsi

   Pages 97-126

8. Integration

   • Asuman G. Aksoy, Mohamed A. Khamsi

   Pages 127-158

9. Series

   • Asuman G. Aksoy, Mohamed A. Khamsi

   Pages 159-180

10. Metric Spaces

    • Asuman G. Aksoy, Mohamed A. Khamsi

    Pages 181-195

11. Fundamentals of Topology

    • Asuman G. Aksoy, Mohamed A. Khamsi

    Pages 197-221

12. Sequences and Series of Functions

    • Asuman G. Aksoy, Mohamed A. Khamsi

    Pages 223-247

13. Back Matter

    Pages 1-6

    PDF

Education is an admirable thing, but it is well to remember from time to time that nothing worth knowing can be taught. Oscar Wilde, “The Critic as Artist,” 1890. Analysis is a profound subject; it is neither easy to understand nor summarize. However, Real Analysis can be discovered by solving problems. This book aims to give independent students the opportunity to discover Real Analysis by themselves through problem solving. ThedepthandcomplexityofthetheoryofAnalysiscanbeappreciatedbytakingaglimpseatits developmental history. Although Analysis was conceived in the 17th century during the Scienti?c Revolution, it has taken nearly two hundred years to establish its theoretical basis. Kepler, Galileo, Descartes, Fermat, Newton and Leibniz were among those who contributed to its genesis. Deep conceptual changes in Analysis were brought about in the 19th century by Cauchy and Weierstrass. Furthermore, modern concepts such as open and closed sets were introduced in the 1900s. Today nearly every undergraduate mathematics program requires at least one semester of Real Analysis. Often, students consider this course to be the most challenging or even intimidating of all their mathematics major requirements. The primary goal of this book is to alleviate those concerns by systematically solving the problems related to the core concepts of most analysis courses. In doing so, we hope that learning analysis becomes less taxing and thereby more satisfying.

• Riemann
• Taylor's theorem
• analysis
• elementary logic
• fundamentals topology
• improper integral
• intermediate value theorem
• limits functions
• linear algebra
• mean value theorem
• real analysis
• sequences functions
• series functions
• set theory
• upper sum

From the reviews:

“Spread out over 11 chapters, this is a collection of 319 problems in what used to be called Advanced Calculus. … The authors see their book primarily as an aid to undergraduates … but I view it as being helpful to teachers in supplementing their courses or in preparing exams. … However, kept on a course reserve shelf of an academic library, the book under review might entice and benefit the more dedicated student. It certainly merits the attention of instructors of elementary analysis.” (Henry Ricardo, The Mathematical Association of America, June, 2010)

“A very readable collection of interesting problems of varying levels of difficulty. It is intended to build a bridge between ordinary high school or undergraduate exercises and more difficult and abstract concepts or problems. The book is so delightfully written that anyone who simply likes working on challenging problems could read it independently. … recommends this book to all students curious about elementary real analysis and how to learn it through solving problems. … a welcome resource for organizing their activities at a good level.” (Vicenţiu D. Rădulescu, Zentralblatt MATH, Vol. 1186, 2010)

• Dept. Mathematics & Computer Science, Claremont McKenna College, Claremont, U.S.A.

  Asuman G. Aksoy

• Dept. Mathematical Sciences, University of Texas, El Paso (UTEP), El Paso, U.S.A.

  Mohamed A. Khamsi

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