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This is a text for a two-term course in introductory real analysis for junior or senior math- ematics majors and science students with a serious interest in mathematics. Prospective educators or mathematically gifted high school students can also benefit from the mathe- matical maturity that can be gained from an introductory real analysis course.

The book is designed to fill the gaps left in the development of calculus as it is usually presented in an elementary course, and to provide the background required for insight into more advanced courses in pure and applied mathematics. The standard elementary calcu- lus sequence is the only specific prerequisite for Chapters 1–5, which deal with real-valued functions. (However, other analysis oriented courses, such as elementary differential equa- tion, also provide useful preparatory experience.) Chapters 6 and 7 require a working knowledge of determinants, matrices and linear transformations, typically available from a first course in linear algebra. Chapter 8 is accessible after completion of Chapters 1–5.

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Pearson Education


Mathematics | Numerical Analysis and Computation


This book has been judged to meet the evaluation criteria set by the Editorial Board of the American Institute of Mathematics in connection with the Institute’s Open Textbook Initiative. It may be copied, modified, redistributed, translated, and built upon sub- ject to the Creative Commons

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Library of Congress Cataloging-in-Publication Data

Trench, William F. Introduction to real analysis / William F. Trench

p. cm. ISBN 0-13-045786-8

1. Mathematical Analysis. QA300.T667 2003

I. Title. 515-dc21 2002032369

Free Hyperlinked Edition 2.04 December 2013

This book was published previously by Pearson Education.

This free edition is made available in the hope that it will be useful as a textbook or refer- ence. Reproduction is permitted for any valid noncommercial educational, mathematical, or scientific purpose. However, charges for profit beyond reasonable printing costs are prohibited.

A complete instructor’s solution manual is available by email to, sub- ject to verification of the requestor’s faculty status. Although this book is subject to a Creative Commons license, the solutions manual is not. The author reserves all rights to the manual.

Introduction to Real Analysis