Real Analysis

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INTRODUCTION TO REAL ANALYSIS William F. Trench Andrew G. Cowles Distinguished Professor Emeritus Department of Mathematics Trinity University San Antonio, Texas, USA 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 subject to the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. FREE DOWNLOADABLE SUPPLEMENTS FUNCTIONS DEFINED BY IMPROPER INTEGRALS THE METHOD OF LAGRANGE MULTIPLIERS 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. I. Title. QA300.T667 2003 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 reference. 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, subject 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. TO BEVERLY Contents Preface Chapter 1 vi The Real Numbers 1 1.1 The Real Number System 1.2 Mathematical Induction 1.3 The Real Line Chapter 2 2.1 2.2 2.3 2.4 2.5 Differential Calculus of Functions of One Variable 30 Functions and Limits Continuity Differentiable Functions of One Variable

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