مسائلی از آنالیز حقیقی

A Problem Book in Real Analysis

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.
The depth and complexity of the theory of Analysis can be appreciated by taking a glimpse at its developmental history. Although Analysis was conceived in the 17th century during the Scientific 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.
The wide variety of exercises presented in this book range from the computational to the more conceptual and vary in difficulty. They cover the following subjects: Set Theory, Real Numbers, Sequences, Limits of Functions, Continuity, Differentiability, Integration, Series, Metric Spaces, Sequences and Series of Functions and Fundamentals of Topology. Prerequisites for accessing this book are a robust understanding of Calculus and Linear Algebra. While we define the concepts and cite theorems used in each chapter, it is best to use this book alongside standard analysis books such as: Principles of Mathematical Analysis by W. Rudin, Understanding Analysis by S.
Abbott, Elementary Classical Analysis by J. E. Marsden and M. J. Hoffman, and Elements of Real Analysis by D. A. Sprecher. A list of analysis texts is provided at the end of the book.
Although A Problem Book in Real Analysis is intended mainly for undergraduate mathematics students, it can also be used by teachers to enhance their lectures or as an aid in preparing exams.
The proper way to use this book is for students to first attempt to solve its problems without looking at solutions. Furthermore, students should try to produce solutions which are different from those presented in this book. It is through the search for a solution that one learns most mathematics.
Knowledge accumulated from many analysis books we have studied in the past has surely influenced the solutions we have given here. Giving proper credit to all the contributors is a difficult task that we have not undertaken; however, they are all appreciated. We also thank Claremont students Aaron J. Arvey, Vincent E. Selhorst-Jones and Martijn van Schaardenburg for their help with LaTeX. The source for the photographs and quotes given at the beginning of each chapter in this book are from the archive at http://www-history.mcs.st-andrews.ac.uk/ Perhaps Oscar Wilde is correct in saying “nothing worth knowing can be taught.” Regardless, teachers can show that there are paths to knowledge. This book is intended to reveal such a path to understanding Real Analysis. A Problem Book in Real Analysis is not simply a collection of problems; it intends to stimulate its readers to independent thought in discovering Analysis.

Mohamed Amine Khamsi

May 2009