Problem Book in Real Analysis

This book systematically solves the problems related to the core concepts of most analysis courses. The wide variety of exercises presented in this book range from the computational to the more conceptual and vary in difficulty.

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 A Problem Book in Real Analysis is to alleviate those concerns by systematically solving the problems related to the core concepts of most analysis courses. In doing so, the authors hope that learning analysis becomes less taxing and more satisfying.The wide variety of exercises presented in this book range from the computational to the more conceptual and varies in difficulty. They cover the following subjects: set theory; real numbers; sequences; limits of the functions; continuity; differentiability; integration; series; metric spaces; sequences; and series of functions and fundamentals of topology. Furthermore, the authors define the concepts and cite the theorems used at the beginning of each chapter. 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.Prerequisites for accessing this book are a robust understanding of calculus and linear algebra.

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 A Problem Book in Real Analysis is to alleviate those concerns by systematically solving the problems related to the core concepts of most analysis courses. In doing so, the authors hope that learning analysis becomes less taxing and more satisfying. The wide variety of exercises presented in this book range from the computational to the more conceptual and varies in difficulty. They cover the following subjects: set theory; real numbers; sequences; limits of the functions; continuity; differentiability; integration; series; metric spaces; sequences; and series of functions and fundamentals of topology. Furthermore, the authors define the concepts and cite the theorems used at the beginning of each chapter. 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. Prerequisites for accessing this book are a robust understanding of calculus and linear algebra.

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." (Vicentiu D. Radulescu, Zentralblatt MATH, Vol. 1186, 2010)

Preface		ix
Elementary Logic and Set Theory		1	(20)
Solutions		9	(12)
Real Numbers		21	(20)
Solutions		27	(14)
Sequences		41	(22)
Solutions		47	(16)
Limits of Functions		63	(14)
Solutions		68	(9)
Continuity		77	(20)
Solutions		84	(13)
Differentiability		97	(30)
Solutions		105	(22)
Integration		127	(32)
Solutions		136	(23)
Series		159	(22)
Solutions		166	(15)
Metric Spaces		181	(16)
Solutions		186	(11)
Fundamentals of Topology		197	(26)
Solutions		206	(17)
Sequences and Series of Functions		223	(26)
Solutions		231	(18)
Bibliography		249	(2)
Index		251