A Course in Calculus and Real Analysis pdf pdf
Undergraduate Texts in Mathematics Editors
S. Axler K.A. Ribet
Undergraduate Texts in Mathematics (continued after index) Abbott: Understanding Analysis.
From Rabbits to Chaos Curtis: Linear Algebra: An Introductory Approach. Fourth edition.
Reflections: In a Room with Many Mirrors.
Harris/Hirst/Mossinghoff: Combinatorics and Graph Theory. Hartshorne: Geometry: Euclid and Beyond. Hijab: Introduction to Calculus and Classical Analysis. Hilton/Holton/Pedersen: Mathematical
Readings in Mathematics.
Halmos: Naive Set Theory. Hämmerlin/Hoffmann: Numerical Mathematics.
Second edition.
Halmos: Finite-Dimensional Vector Spaces.
Readings in Mathematics.
Hairer/Wanner: Analysis by Its History.
Real Analysis Gordon: Discrete Probability.
Foulds: Combinatorial Optimization for Undergraduates. Foulds: Optimization Techniques: An Introduction. Franklin: Methods of Mathematical Economics. Frazier: An Introduction to Wavelets Through Linear Algebra. Gamelin: Complex Analysis. Ghorpade/Limaye: A Course in Calculus and
Second edition.
Edgar: Measure, Topology, and Fractal Geometry. Elaydi: An Introduction to Difference Equations. Third edition. Erdõs/Surányi: Topics in the Theory of Numbers. Estep: Practical Analysis on One Variable. Exner: An Accompaniment to Higher Mathematics. Exner: Inside Calculus. Fine/Rosenberger: The Fundamental Theory of Algebra. Fischer: Intermediate Real Analysis. Flanigan/Kazdan: Calculus Two: Linear and Nonlinear Functions. Second edition. Fleming: Functions of Several Variables.
Logic. Second edition.
Daepp/Gorkin: Reading, Writing, and Proving: A Closer Look at Mathematics. Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory. Second edition. Dixmier: General Topology. Driver: Why Math? Ebbinghaus/Flum/Thomas: Mathematical
Cox/Little/O’Shea: Ideals, Varieties, and Algorithms. Second edition. Croom: Basic Concepts of Algebraic Topology. Cull/Flahive/Robson: Difference Equations.
Anglin: Mathematics: A Concise History and Philosophy.
Fourth edition.
Chung/AitSahlia: Elementary Probability Theory: With Stochastic Processes and an Introduction to Mathematical Finance.
Chambert-Loir: A Field Guide to Algebra Childs: A Concrete Introduction to Higher Algebra. Second edition.
Second edition.
Carter/van Brunt: The Lebesgue– Stieltjes Integral: A Practical Introduction. Cederberg: A Course in Modern Geometries.
Introduction to Special and General Relavitity.
Buskes/van Rooij: Topological Spaces: From Distance to Neighborhood. Callahan: The Geometry of Spacetime: An
Second Edition.
Brickman: Mathematical Introduction to Linear Programming and Game Theory. Browder: Mathematical Analysis: An Introduction. Buchmann: Introduction to Cryptography.
Readings in Mathematics.
Apostol: Introduction to Analytic Number Theory. Second edition. Armstrong: Basic Topology. Armstrong: Groups and Symmetry. Axler: Linear Algebra Done Right. Second edition. Beardon: Limits: A New Approach to Real Analysis. Bak/Newman: Complex Analysis. Second edition. Banchoff/Wermer: Linear Algebra Through Geometry. Second edition. Berberian: A First Course in Real Analysis. Bix: Conics and Cubics: A Concrete Introduction to Algebraic Curves. Brèmaud: An Introduction to Probabilistic Modeling. Bressoud: Factorization and Primality Testing. Bressoud: Second Year Calculus.
Readings in Mathematics.
Anglin/Lambek: The Heritage of Thales.
Readings in Mathematics.
Hilton/Holton/Pedersen: Mathematical Vistas: From a Room with Many Windows. Iooss/Joseph: Elementary Stability and Bifurcation Theory. Second Edition.
Sudhir R. Ghorpade Balmohan V. Limaye
A Course in Calculus and Real Analysis With 71 Figures Sudhir R. Ghorpade Balmohan V. Limaye Department of Mathematics Department of Mathematics Indian Institute of Technology Bombay Indian Institute of Technology Bombay Powai, Mumbai 400076 Powai, Mumbai 400076
INDIA
INDIA srg@math.iitb.ac.in bvl@math.iitb.ac.in
Series Editors:
S. Axler K. A. Ribet Mathematics Department Department of Mathematics San Francisco State University University of California at Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA axler@sfsu.edu ribet@math.berkeley.edu
Mathematics Subject Classification (2000): 26-01 40-XX Library of Congress Control Number: 2006920312
ISBN-10: 0-387-30530-0 Printed on acid-free paper.
ISBN-13: 978-0387-30530-1 © 2006 Springer Science+Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street,
New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly
analysis. Use in connection with any form of information storage and retrieval, electronic adap-
tation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if
they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (MVY) 9 8 7 6 5 4 3 2 1 springer.com Preface
Calculus is one of the triumphs of the human mind. It emerged from inves- tigations into such basic questions as finding areas, lengths and volumes. In the third century B.C., Archimedes determined the area under the arc of a parabola. In the early seventeenth century, Fermat and Descartes studied the problem of finding tangents to curves. But the subject really came to life in the hands of Newton and Leibniz in the late seventeenth century. In partic- ular, they showed that the geometric problems of finding the areas of planar regions and of finding the tangents to plane curves are intimately related to one another. In subsequent decades, the subject developed further through the work of several mathematicians, most notably Euler, Cauchy, Riemann, and Weierstrass.
Today, calculus occupies a central place in mathematics and is an essential component of undergraduate education. It has an immense number of appli- cations both within and outside mathematics. Judged by the sheer variety of the concepts and results it has generated, calculus can be rightly viewed as a fountainhead of ideas and disciplines in mathematics.
Real analysis, often called mathematical analysis or simply analysis, may be regarded as a formidable counterpart of calculus. It is a subject where one revisits notions encountered in calculus, but with greater rigor and sometimes with greater generality. Nonetheless, the basic objects of study remain the same, namely, real-valued functions of one or several real variables.
This book attempts to give a self-contained and rigorous introduction to calculus of functions of one variable. The presentation and sequencing of topics emphasizes the structural development of calculus. At the same time, due im- portance is given to computational techniques and applications. In the course of our exposition, we highlight the fact that calculus provides a firm foun- dation to several concepts and results that are generally encountered in high school and accepted on faith. For instance, this book can help students get a clear understanding of (i) the definitions of the logarithmic, exponential and trigonometric functions and a proof of the fact that these are not algebraic functions, (ii) the definition of an angle and (iii) the result that the ratio of VI Preface
the circumference of a circle to its diameter is the same for all circles. It is our experience that a majority of students are unable to absorb these concepts and results without getting into vicious circles. This may partly be due to the division of calculus and real analysis in compartmentalized courses. Calculus is often taught as a service course and as such there is little time to dwell on subtleties and gain perspective. On the other hand, real analysis courses may start at once with metric spaces and devote more time to pathological exam- ples than to consolidating students’ knowledge of calculus. A host of topics such as L’Hˆ opital’s rule, points of inflection, convergence criteria for Newton’s method, solids of revolution, and quadrature rules, which may have been in- adequately covered in calculus courses, become pass´e when one studies real analysis. Trigonometric, exponential, and logarithmic functions are defined, if at all, in terms of infinite series, thereby missing out on purely algebraic moti- vations for introducing these functions. The ubiquitous role of π as a ratio of various geometric quantities and as a constant that can be defined indepen- dently using calculus is often not well understood. A possible remedy would be to avoid the separation of calculus and real analysis into seemingly disjoint courses and textbooks. Attempts along these lines have been made in the past as in the excellent books of Hardy and of Courant and John. Ours is another attempt to give a unified exposition of calculus and real analysis and address the concerns expressed above. While this book deals with functions of one variable, we intend to treat functions of several variables in another book.
The genesis of this book lies in the notes we prepared for an undergraduate course at the Indian Institute of Technology Bombay in 1997. Encouraged by the feedback from students and colleagues, the notes and problem sets were put together in March 1998 into a booklet that has been in private circulation. Initially, it seemed that it would be relatively easy to convert that booklet into a book. Seven years have passed since then and we now know a little better! While that booklet was certainly helpful, this book has evolved to acquire a form and philosophy of its own and is quite distinct from the original notes.
A glance at the table of contents should give the reader an idea of the topics covered. For the most part, these are standard topics and novelty, if any, lies in how we approach them. Throughout this text we have sought to make a distinction between the intrinsic definition of a geometric notion and the analytic characterizations or criteria that are normally employed in studying it. In many cases we have used articles such as those in A Century of Calculus to simplify the treatment. Usually each important result is followed by two kinds of examples: one to illustrate the result and the other to show that a hypothesis cannot be dropped.
When a concept is defined it appears in boldface. Definitions are not num- bered but can be located using the index. Everything else (propositions, ex- amples, remarks, etc.) is numbered serially in each chapter. The end of a proof is marked by the symbol
⊔, while the symbol ✸ marks the end of an example ⊓ or a remark. Bibliographic details about the books and articles mentioned in the text and in this preface can be found in the list of references. Citations Preface
VII
within the text appear in square brackets. A list of symbols and abbreviations used in the text appears after the list of references.
The Notes and Comments that appear at the end of each chapter are an important part of the book. Distinctive features of the exposition are men- tioned here and often pointers to some relevant literature and further devel- opments are provided. We hope that these may inspire many fruitful visits to the library—not when a quiz or the final is around the corner, but per- haps after it is over. The Notes and Comments are followed by a fairly large collection of exercises. These are divided into two parts. Exercises in Part A are relatively routine and should be attempted by all students. Part B con- tains problems that are of a theoretical nature or are particularly challenging. These may be done at leisure. Besides the two sets of exercises in every chap- ter, there is a separate collection of problems, called Revision Exercises which appear at the end of Chapter 7. It is in Chapter 7 that the logarithmic, ex- ponential, and trigonometric functions are formally introduced. Their use is strictly avoided in the preceding chapters. This meant that standard examples and counterexamples such as x sin(1/x) could not be discussed earlier. The Revision Exercises provide an opportunity to revisit the material covered in Chapters 1–7 and to work out problems that involve the use of elementary transcendental functions.
The formal prerequisites for this course do not go beyond what is normally covered in high school. No knowledge of trigonometry is assumed and expo- sure to linear algebra is not taken for granted. However, we do expect some mathematical maturity and an ability to understand and appreciate proofs. This book can be used as a textbook for a serious undergraduate course in calculus. Parts of the book could be useful for advanced undergraduate and graduate courses in real analysis. Further, this book can also be used for self- study by students who wish to consolidate their knowledge of calculus and real analysis. For teachers and researchers this may be a useful reference for topics that are usually not covered in standard texts.
Apart from the first paragraph of this preface, we have not discussed the history of the subject or placed each result in historical context. However, we do not doubt that a knowledge of the historical development of concepts and results is important as well as interesting. Indeed, it can greatly enrich one’s understanding and appreciation of the subject. For those interested, we encourage looking on the Internet, where a wealth of information about the history of mathematics and mathematicians can be readily found. Among the various sources available, we particularly recommend the MacTutor History of Mathematics archive http://www-groups.dcs.st-and.ac.uk/history/ at the University of St. Andrews. The books of Boyer, Edwards, and Stillwell are also excellent sources for the history of mathematics, especially calculus.
In preparing this book we have received generous assistance from vari- ous organizations and individuals. First, we thank our parent institution IIT Bombay and in particular its Department of Mathematics for providing ex- cellent infrastructure and granting a sabbatical leave for each of us to work VIII Preface
on this book. Financial assistance for the preparation of this book was re- ceived from the Curriculum Development Cell at IIT Bombay, for which we are thankful. Several colleagues and students have read parts of this book and have pointed out errors in earlier versions and made a number of useful suggestions. We are indebted to all of them and we mention, in particular, Rafikul Alam, Swanand Khare, Rekha P. Kulkarni, Narayanan Namboodri, S. H. Patil, Tony J. Puthenpurakal, P. Shunmugaraj, and Gopal K. Srinivasan. The figures in the book have been drawn using PSTricks, and this is the work of Habeeb Basha and to a greater extent of Arunkumar Patil. We appreciate their efforts, and are grateful to them. Thanks are also due to C. L. Anthony, who typed a substantial part of the manuscript. The editorial and TeXnical staff at Springer, New York, have always been helpful and we thank all of them, especially Ina Lindemann and Mark Spencer for believing in us and for their patience and cooperation. We are also grateful to David Kramer, who did an excellent job of copyediting and provided sound counsel on linguistic and stylistic matters. We owe more than gratitude to Sharmila Ghorpade and Nirmala Limaye for their support.
We would appreciate receiving comments, suggestions, and corrections. These can be sent by e-mail to acicara@gmail.com or by writing to either of us. Corrections, modifications, and relevant information will be posted at http://www.math.iitb.ac.in/
∼srg/acicara and we encourage interested readers to visit this website to learn about updates concerning the book. Mumbai, India Sudhir Ghorpade July 2005
Balmohan Limaye Contents 1 Numbers and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1.1 Properties of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Functions and Their Geometric Properties . . . . . . . . . . . . . . . . . . 13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1 Convergence of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Subsequences and Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . 55 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3 Continuity and Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.1 Continuity of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Basic Properties of Continuous Functions . . . . . . . . . . . . . . . . . . . 72
3.3 Limits of Functions of a Real Variable . . . . . . . . . . . . . . . . . . . . . . 81 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1 The Derivative and Its Basic Properties . . . . . . . . . . . . . . . . . . . . 104
4.2 The Mean Value and Taylor Theorems . . . . . . . . . . . . . . . . . . . . . 117
4.3 Monotonicity, Convexity, and Concavity . . . . . . . . . . . . . . . . . . . . 125
4.4 L’Hˆopital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5 Applications of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.1 Absolute Minimum and Maximum . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.2 Local Extrema and Points of Inflection . . . . . . . . . . . . . . . . . . . . . 150
5.3 Linear and Quadratic Approximations . . . . . . . . . . . . . . . . . . . . . 157
5.4 The Picard and Newton Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 X Contents
6 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.1 The Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.2 Integrable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.3 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . 200
6.4 Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
7 Elementary Transcendental Functions . . . . . . . . . . . . . . . . . . . . . 227
7.1 Logarithmic and Exponential Functions . . . . . . . . . . . . . . . . . . . . 228
7.2 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
7.3 Sine of the Reciprocal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
7.4 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
7.5 Transcendence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Revision Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
8 Applications and Approximations of Riemann Integrals . . . . 291
8.1 Area of a Region Between Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 291
8.2 Volume of a Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
8.3 Arc Length of a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
8.4 Area of a Surface of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
8.5 Centroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
8.6 Quadrature Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
9 Infinite Series and Improper Integrals . . . . . . . . . . . . . . . . . . . . . 361
9.1 Convergence of Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
9.2 Convergence Tests for Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
9.3 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
9.4 Convergence of Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 384
9.5 Convergence Tests for Improper Integrals . . . . . . . . . . . . . . . . . . . 392
9.6 Related Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
1 Numbers and Functions
Let us begin at the beginning. When we learn the script of a language, such as the English language, we begin with the letters of the alphabet A, B, C, . . .; when we learn the sounds of music, such as those of western classical music, we begin with the notes Do, Re, Mi, . . . . Likewise, in mathematics, one begins with 1, 2, 3, . . .; these are the positive integers or the natural numbers. We shall denote the set of positive integers by N. Thus,
N = {1, 2, 3, . . .} . These numbers have been known since antiquity. Over the years, the number 0
1
was conceived and subsequently, the negative integers. Together, these form
2
the set Z of integers. Thus, Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .} . Quotients of integers are called rational numbers. We shall denote the set of all rational numbers by Q. Thus, m Q = : m, n .
∈ Z, n = 0 n Geometrically, the integers can be represented by points on a line by fixing a base point (signifying the number 0) and a unit distance. Such a line is called the number line and it may be drawn as in Figure 1.1. By suitably subdi- viding the segment between 0 and 1, we can also represent rational numbers such as 1/n, where n 1 ∈ N, and this can, in turn, be used to represent any
The invention of ‘zero’, which also paves the way for the place value system of enumeration, is widely credited to the Indians. Great psychological barriers had to be overcome when ‘zero’ was being given the status of a legitimate number. 2 For more on this, see the books of Kaplan [39] and Kline [41].
The notation Z for the set of integers is inspired by the German word Zahlen for numbers.
2
1 Numbers and Functions Fig. 1.1.
The number line
rational number by a unique point on the number line. It is seen that the rational numbers spread themselves rather densely on this line. Nevertheless, √ several gaps do remain. For example, the ‘number’ 2 can be represented by a unique point between 1 and 2 on the number line using simple geometric constructions, but as we shall see later, this is not a rational number. We are, therefore, forced to reckon with the so-called irrational numbers, which are precisely the ‘numbers’ needed to fill the gaps left on the number line after marking all the rational numbers. The rational numbers and the irrational numbers together constitute the set R, called the set of real numbers. The geometric representation of the real numbers as points on the number line naturally implies that there is an order among the real numbers. In particu- lar, those real numbers that are greater than 0, that is, which correspond to points to the right of 0, are called positive.
1.1 Properties of Real Numbers
To be sure, we haven’t precisely defined what real numbers are and what it means for them to be positive. For that matter, we haven’t even defined
3
the positive integers 1, 2, 3, . . . or the rational numbers. But at least we are familiar with the latter. We are also familiar with the addition and the mul- tiplication of rational numbers. As for the real numbers, which are not easy to define, it is better to at least specify the properties that we shall take for granted. We shall take adequate care that in the subsequent development, we use only these properties or the consequences derived from them. In this way, we don’t end up taking too many things on faith. So let us specify our assumptions.
We assume that there is a set R (whose elements are called real num- bers), which contains the set Q of all rational numbers (and, in particular, the numbers 0 and 1) such that the following three types of properties are satisfied. 3 To a purist, this may appear unsatisfactory. A conscientious beginner in calculus
may also become uncomfortable at some point of time that the basic notion of
a (real) number is undefined. Such persons are first recommended to read the
‘Notes and Comments’ at the end of this chapter and then look up some of the
references mentioned therein.1.1 Properties of Real Numbers
3 Algebraic Properties
We have the operations of addition (denoted by +) and multiplication (de- noted by · or by juxtaposition) on R, which extend the usual addition and multiplication of rational numbers and satisfy the following properties:
A1 a + (b + c) = (a + b) + c and a(bc) = (ab)c for all a, b, c ∈ R. A2 a + b = b + a and ab = ba for all a, b ∈ R. A3 a + 0 = a and a · 1 = a for all a ∈ R.
′ ′
A4 Given any a = 0. Further, if a ∈ R, there is a ∈ R such that a + a = 0,
∗ ∗ then there is a = 1.
∈ R such that aa A5 a(b + c) = ab + ac for all a, b, c ∈ R.
It is interesting to note that several simple properties of real numbers that one is tempted to take for granted can be derived as consequences of the above properties. For example, let us prove that a
· 0 = 0 for all a ∈ R. First, by A3, we have 0 + 0 = 0. So, by A5, a · 0 = a(0 + 0) = a · 0 + a · 0. Now, by A4, there
′ ′
is a b = 0. Thus, ∈ R such that a · 0 + b
′ ′ ′
0 = a = (a = a ) = a · 0 + b · 0 + a · 0) + b · 0 + (a · 0 + b · 0 + 0 = a · 0, where the third equality follows from A1 and the last equality follows from
A3. This completes the proof! A number of similar properties are listed in the exercises and we invite the reader to supply the proofs. These show, in
′ ′
particular, that given any a = 0 is ∈ R, an element a ∈ R such that a + a unique; this element will be called the negative or the additive inverse of
∗
a and denoted by −a. Likewise, if a ∈ R and a = 0, then an element a ∈ R
∗
such that aa = 1 is unique; this element is called the reciprocal or the
−1
multiplicative inverse of a and is denoted by a or by 1/a. Once all these formalities are understood, we will be free to replace expressions such as a (1/b) , a + a, aa, (a + b) + c, (ab)c, a + (
−b), by the corresponding simpler expressions, namely,
2
a/b, 2a, a , a + b + c, abc, a − b. Here, for instance, it is meaningful and unambiguous to write a + b + c, thanks to A1. More generally, given finitely many real numbers a , . . . , a , the sum
1 n
- a has an unambiguous meaning. To represent such sums, the
1 n
· · · + a
- “sigma notation” can be quite useful. Thus, a is often denoted by
1 n
· · · + a
n
a or sometimes simply by a or a . Likewise, the product a
i i i 1 n
i=1 i · · · a
of the real numbers a , . . . , a has an unambiguous meaning and it is often
1 n n
denoted by a or sometimes simply by a or a . We remark that as
i i i
i=1 ia convention, the empty sum is defined to be zero, whereas an empty product
n n is defined to be one. Thus, if n = 0, then a := 0, whereas a := 1. i i i=1 i=1
4
1 Numbers and Functions Order Properties
- The set R contains a subset R , called the set of all positive real numbers, satisfying the following properties: O1 Given any a
∈ R, exactly one of the following statements is true:
- a ; a = 0; .
∈ R −a ∈ R
- O2 If a, b , then a + b and ab .
∈ R ∈ R ∈ R
- Given the existence of R , we can define an order relation on R as follows.
- For a, b .
∈ R, define a to be less than b, and write a < b, if b − a ∈ R Sometimes, we write b > a in place of a < b and say that b is greater than
- a. With this notation, it follows from the algebraic properties that R = {x ∈ R : x > 0}. Moreover, the following properties are easy consequences of A1–A5 and O1–O2:
(i) Given any a, b ∈ R, exactly one of the following statements is true. a < b; a = b; b < a. (ii) If a, b, c ∈ R with a < b and b < c, then a < c. (iii) If a, b, c
∈ R, with a < b, then a+c < b+c. Further, if c > 0, then ac < bc, whereas if c < 0, then ac > bc. Note that it is also a consequence of the properties above that 1 > 0. Indeed, by (i), we have either 1 > 0 or 1 < 0. If we had 1 < 0, then we must have −1 > 0 and hence by (iii), 1 = (−1)(−1) > 0, which is a contradiction.
2 Therefore, 1 > 0. A similar argument shows that a > 0 for any a ∈ R, a = 0.
The notation a ≤ b is often used to mean that either a < b or a = b. Likewise, a ≥ b means that a > b or a = b.
Let S be a subset of R. We say that S is bounded above if there exists α
∈ R such that x ≤ α for all x ∈ S. Any such α is called an upper bound of S. We say that S is bounded below if there exists β ∈ R such that x ≥ β for all x
∈ S. Any such β is called a lower bound of S. The set S is said to be bounded if it is bounded above as well as bounded below; otherwise, S is said to be unbounded. Note that if S =
∅, that is, if S is the empty set, then every real number is an upper bound as well as a lower bound of S. Examples 1.1. (i) The set N of positive integers is bounded below, and any real number β
≤ 1 is a lower bound of N. However, as we shall see later in Proposition 1.3, the set N is not bounded above. (ii) The set S of reciprocals of positive integers, that is,
1
1 S := 1, , , . . .
2
3 is bounded. Any real number α ≥ 1 is an upper bound of S, whereas any real number β
1.1 Properties of Real Numbers
5
2
(iii) The set S := < 2 {x ∈ Q : x } is bounded. Here, for example, 2 is an upper bound, while
✸ −2 is a lower bound. Let S be a subset of R. An element M
∈ R is called a supremum or a least upper bound of the set S if (i) M is an upper bound of S, that is, x
≤ M for all x ∈ S, and (ii) M ≤ α for any upper bound α of S.
It is easy to see from the definition that if S has a supremum, then it must be unique; we denote it by sup S. Note that ∅ does not have a supremum. An element m
∈ R is called an infimum or a greatest lower bound of the set S if (i) m is a lower bound of S, that is, m
≤ x for all x ∈ S, and (ii) β ≤ m for any lower bound β of S.
Again, it is easy to see from the definition that if S has an infimum, then it must be unique; we denote it by inf S. Note that ∅ does not have an infimum. For example, if S =
{x ∈ R : 0 < x ≤ 1}, then inf S = 0 and sup S = 1. In this example, inf S is not an element of S, but sup S is an element of S. If the supremum of a set S is an element of S, then it is called the maxi- mum of S, and denoted by max S; likewise, if the infimum of S is in S, then it is called the minimum of S, and denoted by min S.
The last, and perhaps the most important, property of R that we shall assume is the following.
Completeness Property Every nonempty subset of R that is bounded above has a supremum.
The significance of the Completeness Property (which is also known as the Least Upper Bound Property) will become clearer from the results proved in this as well as the subsequent chapters.
Proposition 1.2. Let S be a nonempty subset of R that is bounded below. Then S has an infimum.
Proof. Let T = {β ∈ R : β ≤ a for all a ∈ S}. Since S is bounded below, T is nonempty, and since S is nonempty, T is bounded above. Hence T has a supremum. It is easily seen that sup T is the infimum of S.
⊓ ⊔ Proposition 1.3. Given any x ∈ R, there is some n ∈ N such that n > x. Consequently, there is also an m ∈ N such that −m < x. Proof. Assume the contrary. Then x is an upper bound of N. Therefore, N has a supremum. Let M = sup N. Then M
− 1 < M and hence M − 1 is not an upper bound of N. So, there is n ∈ N such that M −1 < n. But then n+1 ∈ N and M < n + 1, which is a contradiction since M is an upper bound of N.
The second assertion about the existence of m ∈ N with −m < x follows by applying the first assertion to
−x. ⊔ ⊓
6
1 Numbers and Functions
The first assertion in the proposition above is sometimes referred to as the Archimedean property of R. Observe that for any positive real number ǫ, by applying the Proposition 1.3 to x = 1/ǫ, we see that there exists n
∈ N such that (1/n) < ǫ. Note also that thanks to Proposition 1.3, for any x ∈ R, there are m, n
∈ N such that −m < x < n. The largest among the finitely many integers k satisfying −m ≤ k ≤ n and also k ≤ x is called the integer part of x and is denoted by [x]. Note that the integer part of x is characterized by the following properties:
[x] ∈ Z and [x] ≤ x < [x] + 1. Sometimes, the integer part of x is called the floor of x and is denoted by ⌊x⌋. In the same vein, the smallest integer ≥ x is called the ceiling of x and
3
3
is denoted by = ⌈x⌉. For example, ⌊ ⌋ = ⌊1⌋ = 1, whereas ⌈2⌉ = 2.
2
2 n
Given any a of a to be
∈ R and n ∈ N, we define the nth power a the product a = 1
· · · a of a with itself taken n times. Further, we define a
n −n
and a = (1/a) provided a = 0. In this way integral powers of all nonzero real numbers are defined. The following elementary properties are immediate consequences of the algebraic properties and the order properties of R.
n n n
(i) (a
1 a 2 ) = a a for all n 1 , a
2 1 a
2
1 2 ∈ Z and a ∈ R (with a = 0 if n ≤ 0). n m mn m+n m n
(ii) (a ) = a and a = a a for all m, n ∈ Z and a ∈ R (with a = 0 if m
≤ 0 or n ≤ 0).
n n
(iii) If n
1 , b
2 1 < b 2 , then b < b .
∈ N and b ∈ R with 0 ≤ b
1
2 The first two properties above are sometimes called the laws of exponents or the laws of indices (for integral powers).
Proposition 1.4. Given any n ∈ N and a ∈ R with a ≥ 0, there exists a
n unique b = a.
∈ R such that b ≥ 0 and b Proof. Uniqueness is clear since b , b < b implies that
1
2
1
2
∈ R with 0 ≤ b
n n n
b < b . To prove the existence of b = a, note that the
1 2 ∈ R with b ≥ 0 and b
case of a = 0 is trivial, and moreover, the case of 0 < a < 1 follows from the case of a > 1 by taking reciprocals. Thus we will assume that a ≥ 1. Let
n
S =
a {x ∈ R : x ≤ a}.
Then S is a subset of R, which is nonempty (since 1 ) and bounded
a a
∈ S above (by a, for example). Define b = sup S . Note that since 1 , we have
a a
∈ S
n
b = a by showing that each of the inequalities ≥ 1 > 0. We will show that b
n n b < a and b > a leads to a contradiction.
Note that by Binomial Theorem, for any δ ∈ R, we have n n
n n 2 n n−1 n−2
(b + δ) + = b b δ + + b δ .
· · · + δ
1
2
1.1 Properties of Real Numbers
7 n
Now, suppose b < a. Let us define n ǫ
n n−k
ǫ := a , M := max b : k = 1, . . . , n and δ := min 1, .
−b k nM
k
Then M ≥ 1 and 0 < δ ≤ δ for k = 1, 2, . . . , n. Therefore,
n n 2 n n n + (b + δ) + M δ + M δ + nM δ + ǫ = a.
≤ b · · · + Mδ ≤ b ≤ b Hence, b + δ . But this is a contradiction since b is an upper bound of S .
a a
∈ S
n n
Next, suppose b > a. This time, take ǫ = b − a and define M and δ as before. Similar arguments will show that
n n n
(b − δ) ≥ b − nMδ ≥ b − ǫ = a. But b
. This means
a
− δ < b, and hence b − δ cannot be an upper bound of S
n n
that there is some x such that b < x
a
∈ S − δ < x. Therefore, (b − δ) ≤ a,
n which is a contradiction. Thus b = a.
⊔ ⊓ Thanks to Proposition 1.4, we define, for any n
∈ N and a ∈ R with a ≥ 0,
n
the nth root of a to be the unique real number b such that b = a; ≥ 0 and b
√ n
1/n
we denote this real number by a or by a . In case n = 2, we simply √ √ 2 write a instead of a. From the uniqueness of the nth root, the analogues of the properties (i), (ii), and (iii) stated just before Proposition 1.4 can be easily proved for nth roots instead of the nth powers. More generally, given any
1/n r m
r = (a )
∈ Q, we write r = m/n, where m, n ∈ Z with n > 0, and define a for any a ∈ R with a > 0. Note that if also r = p/q, for some p, q ∈ Z with
1/n 1/q m p
q > 0, then for any a ) = (a ) . This ∈ R with a > 0, we have (a can be seen, for example, by raising both sides to the nqth power, using laws of exponents for integral powers and the uniqueness of roots. Thus, rational powers of positive real numbers are unambiguously defined. In general, for negative real numbers, nonintegral rational powers are not defined in R. For
1/2
2
example, ( cannot equal any b −1) ∈ R since b ≥ 0. However, in a special case, rational powers of negative real numbers can be defined. More precisely, if n
∈ N is odd and a ∈ R is positive, then we define
1/n 1/n
( = a .−a) − It is easily seen that this is well defined, and as a result, for any x
∈ R,
r
x is defined whenever r = 0, the rth power x ∈ Q has an odd denominator, that is, when r = m/n for some m
∈ Z and n ∈ N with n odd. Finally, if
r
r is any positive rational number, then we set 0 = 0. For rational powers, wherever they are defined, analogues of the properties (i), (ii), and (iii) stated just before Proposition 1.4 are valid. These analogues can be easily proved by raising both sides of the desired equality or inequality to sufficiently high integral powers so as to reduce to the corresponding properties of integral powers, and using the uniqueness of roots.
8
1 Numbers and Functions
Real numbers that are not rational numbers are called irrational num- bers. The possibility of taking nth roots provides a useful method to con- struct several examples of irrational numbers. For instance, we prove below a
√ classical result that 2 is an irrational number. The proof here is such that it can easily be adapted to prove that several such numbers, for example, √ √ √ 3 √ 5
3, 15, 2, 16, are not rational. [See Exercises 10 and 44.] We recall first the familiar notion of divisibility in the set Z of integers. Given m, n ∈ Z, we say that m divides n or that m is a factor of n (and write m
| n) if n = ℓm for some ℓ ∈ Z. Sometimes, we write m ∤ n if m does not divide n. Two integers m and n are said to be relatively prime if the only integers
′
that divide both m and n are 1 and −1. It can be shown that if m, n, n ∈ Z
′ ′
are such that m, n are relatively prime and m , then m . It can also | nn | n be shown that any rational number r can be written as p r = , where p, q
∈ Z, q > 0, and p, q are relatively prime. q
The above representation of r is called the reduced form of r. The numerator (namely, p) and the denominator (namely, q) in the case of a reduced form representation are uniquely determined by r.
Proposition 1.5. No rational number has a square equal to 2. In other words, √ 2 is an irrational number.
√ √ Proof. Suppose 2 is rational. Write 2 in the reduced form as p/q, where
2
2
p, q = 2q . Hence q divides
∈ Z, q > 0, and p, q are relatively prime. Then p
2
p . This implies that q divides p, and so p/q is an integer. But there is no
2
integer whose square is 2 because ( = 1 and the square of any integer ±1)
√ other than 1 or 2 is not rational. −1 is ≥ 4. Hence ⊓ ⊔
The following result shows that the rational numbers as well as the irra- tional numbers spread themselves rather densely on the number line. Proposition 1.6. Given any a, b
∈ R with a < b, there exists a rational number as well as an irrational number between a and b. Proof. By Proposition 1.3, we can find n
∈ N such that n > 1/(b − a). Let m = [na] + 1. Then m − 1 ≤ na < m, and hence m na + 1