Introduction to Mathematical Analysis

374 Pages·1982·24.137 MB·English

by  William R Parzynski and Philip W. Zipse

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WILLIAM R.PARZYNSKI PHILIP W. ZIPSE
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INTRODUCTION TO
MATHEMATICAL
ANALYSIS
International Series in Pure and Applied Mathematics
G. Springer
Consulting Editor
Ahifors: Complex Analysis
Bender and Orszag: Advanced Mathematical Methodsfor Scientists and Engineers
Buck: Advanced Calculus
Busacker and Saaty: Finite Graphs and Networks
Cheney: Introduction to Approximation Theory
Chester: Techniques in Partial Differential Equations
Coddington and Levinson: Theory ofOrdinary Differential Equations
Conte and de Boor: Elementary Numerical Analysis: An Algorithmic Approach
Dennemeyer: Introduction to Partial Differential Equations and
Boundary Value Problems
Dettman: Mathematical Methods in Physics and Engineering
Hamming: Numerical Methodsfor Scientists and Engineers
Hildebrand: Introduction to Numerical Analysis
Householder: The Numerical Treatment ofa Single Nonlinear Equation
Kalman, Falb, and Arbib: Topics in Mathematical Systems Theory
McCarty: Topology: An Introduction with Applications to Topological Groups
Moore: Elements ofLinear Algebra and Matrix Theory
Moursund and Duris: Elementary Theory and Application ofNumerical Analysis
Parzynski and Zipse: Introduction to Mathematical Analysis
Pipes and Harvill: Applied Mathematicsfor Engineers and Physicists
Ralston and Rabinowitz: A First Course in Numerical Analysis
Ritger and Rose: Differential Equations with Applications
Rudin: Principles ofMathematical Analysis
Shapiro: Introduction to Abstract Algebra
Simmons: Differential Equations with Applications and Historical Notes
Simmons: Introduction to Topology and Modern Analysis
Struble: Nonlinear Differential Equations
INTRODUCTION TO
MATHEMATICAL
ANALYSIS
William R. Parzynski
Philip W. Zipse
Department ofMathematics andComputer Science
Montclair State College
McGraw-Hill Book Company
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INTRODUCTION TO MATHEMATICAL ANALYSIS
Copyright © 1982 by McGraw-Hill, Inc. All rights reserved.
Printed in the United StatesofAmerica. Except as permitted underthe United
StatesCopyrightAct of 1976, no part ofthispublication may bereproduced or
distributed in any form orby anymeans, orstored inadata base orretrieval
system, without the priorwritten permission ofthe publisher.
34567890 DODO 8987
ISBN -07-0^fl5^S-^
LibraryofCongressCatalogingin Publication Data
Parzynski, William R.
Introduction to mathematical analysis.
(International series in pureand applied mathematics)
Includes index.
1. Mathematical analysis. I. Zipse, Philip W.
II. Title. III. Series.
QA300.P36 515 81-19304
ISBN 0-07-048845-2 AACR2
CONTENTS
Preface
vii
Chapter 1 Real Numbers and Functions
1
i.i Sets and Functions 1
*1.2 The Natural Numbers 6
*1.3 Development ofRational Numbers 11
*1.4 Construction ofthe Real Number System 19
1.5 Properties ofReal Numbers 26
1.6 Functions and One-to-One Correspondences 29
Chapter 2 Sequences and Sets of Real Numbers 37
2.1 Limits ofSequences 37
2.2 Bounded Sequences 45
2.3 Sets ofReal Numbers 53
Chapter 3 Functions and Limits 59
3.1 Bounded Functions 59
3.2 Limits ofFunctions 65
3.3 One-Sided Limits, Infinite Limits, and Limits at Infinity 77
3.4 Monotone Functions 89
Chapter 4 Continuous Functions 94
4.1 Continuity 94
4.2 Properties ofContinuous Functions 102
4.3 Uniform Continuity no
*4.4 Further Topics on Continuity 117
Optional sections.
CONTENTS
vi
Chapter 5 Differentiable Functions 126
5.1 The Derivative 126
5.2 Properties ofDifferentiable Functions 131
5.3 L’Hospital’s Rule 138
*5.4 Further Topics on Differentiation 144
Chapter 6 The Riemann Integral 149
6.1 Definition ofthe Integral 149
6.2 Properties ofthe Integral 162
6.3 Fundamental Theorem ofCalculus 172
*6.4 Necessary and Sufficient Conditions for
Riemann Integrability 178
Chapter 7 Sequences and Series of Functions 184
7.1 Infinite Series ofReal Numbers 184
7.2 Pointwise and Uniform Convergence 195
7.3 Importance ofUniform Convergence 201
7.4 Power Series and Taylor Series 208
Chapter 8 Differentiable Functions of
Several Variables 217
8.1 Sets and Functions in R” 217
8.2 Differentiable Functions 232
8.3 Chain Rules and Taylor’s Formula 239
8.4 Implicit- and Inverse-Function Theorems 246
8.5 Extrema ofFunctions ofSeveral Variables 254
Chapter 9 Multiple Integrals 261
9.1 The Double Integral 261
9.2 Evaluation ofDouble Integrals 271
9.3 Change ofVariables in an Integral 281
9.4 Improper Integrals 286
Chapter 10 Metric Spaces 297
10.1 Definition and Examples 297
10.2 Open and Closed Sets; Topology 302
10.3 Convergence and Completeness 308
10.4 Continuity and Compactness 315
10.5 Applications 324
Solutions and Hints to Selected Exercises 330
Index
353
PREFACE
This book has been written to serve as a text for students in a first course in
mathematical analysis. Such a course would usually follow rather quickly
the traditional freshman calculus and generally appears in college catalogs
under the name of Introduction to Analysis, Mathematical Analysis, or
Advanced Calculus. The body of material which has come to be known as
freshman calculus serves as a prerequisite. This presumes that the reader is
equipped with good skills in advanced high school or college algebra and in
trigonometry.
Today calculus serves a much more diverse audience than in years
past. Not only are mathematics and natural science majors and engineering
students taking calculus but there are growing numbers of business, sta-
tistics, and computer science students in these courses. Traditionally, the
purpose of freshman calculus has been to teach the student the facts and
applications of calculus. This means that the student acquires skill in the
mechanics of calculus and a certain level of proficiency at using calculus in
those many areas where the methods of calculus prove useful. Often, how-
ever, little time in the freshman calculus course is spent on the theory which
enhances one’s understanding of the subject. This is especially true today,
when the trend has been to expose the student as early as possible to a
widening variety of calculus applications. It is the role of the sophomore or
junior level mathematical analysis or advanced calculus course to provide
the understanding that is so often lacking when the student leaves the
freshman calculus course.
The authors believe that the purpose of advanced calculus is twofold:
(1) to allow the student to become acquainted with, and develop a certain
level ofproficiency in, the techniques and methods ofmathematical analysis
(sometimes the proof is more important than the theorem) and (2) to be
VII