Elementary Real Analysis

by Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M.

• ISBN13: 9780130190758
• ISBN10: 0130190756
• Edition: 1st
• Format: Hardcover
• Copyright: 2001-01-01
• Publisher: Pearson College Div

Summary

Elementary Real Analysis is written in a rigorous, yet reader friendly style with motivational and historical material that emphasizes the "big picture" and makes proofs seem natural rather than mysterious. Introduces key concepts such as point set theory, uniform continuity of functions and uniform convergence of sequences of functions. Covers metric spaces. Ideal for readers interested in mathematics, particularly in advanced calculus and real analysis.

Table of Contents

Preface	xi
Properties of the Real Numbers	1	(22)
Introduction	1	(1)
The Real Number System	2	(3)
Algebraic Structure	5	(3)
Order Structure	8	(1)
Bounds	9	(1)
Sups and Infs	10	(3)
The Archimedean Property	13	(2)
Inductive Property of IN	15	(1)
The Rational Numbers Are Dense	16	(2)
The Metric Structure of R	18	(3)
Challenging Problems for Chapter 1	21	(2)
Sequences	23	(54)
Introduction	23	(2)
Sequences	25	(4)
Sequence Examples	26	(3)
†Countable Sets	29	(3)
Convergence	32	(5)
Divergence	37	(2)
Boundedness Properties of Limits	39	(2)
Algebra of Limits	41	(6)
Order Properties of Limits	47	(5)
Monotone Convergence Criterion	52	(4)
Examples of Limits	56	(5)
Subsequences	61	(4)
Cauchy Convergence Criterion	65	(3)
*Upper and Lower Limits	68	(6)
Challenging Problems for Chapter 2	74	(3)
Infinite Sums	77	(81)
Introduction	77	(1)
Finite Sums	78	(6)
*Infinite Unordered sums	84	(6)
*Cauchy Criterion	86	(4)
Ordered Sums: Series	90	(8)
Properties	91	(1)
Special Series	92	(6)
Criteria for Convergence	98	(6)
Boundedness Criterion	99	(1)
Cauchy Criterion	99	(1)
Absolute Convergence	100	(4)
Tests for Convergence	104	(25)
Trivial Test	104	(1)
Direct Comparison Tests	105	(2)
Limit Comparison Tests	107	(1)
Ratio Comparison Test	108	(1)
d'Alembert's Ratio Test	109	(2)
Cauchy's Root Test	111	(1)
†Cauchy's Condensation Test	112	(2)
†Integral Test	114	(1)
*Kummer's Tests	115	(3)
*Raabe's Ratio Test	118	(1)
*Gauss's Ratio Test	118	(3)
Alternating Series Test	121	(1)
*Dirichlet's Test	122	(1)
*Abel's Test	123	(6)
†Rearrangements	129	(6)
Unconditional Convergence	130	(1)
Conditional Convergence	131	(2)
*Comparison of Σi∞=1 ai and Σi&epsis;IN ai	133	(2)
†Products of Series	135	(6)
Products of Absolutely Convergent Series	138	(1)
Products of Nonabsolutely Convergent Series	139	(2)
*Summability Methods	141	(7)
*Cesaro's Method	142	(2)
*Abel's Method	144	(4)
*More on Infinite Sums	148	(2)
†Infinite Products	150	(4)
Challenging Problems for Chapter 3	154	(4)
Sets of Real Numbers	158	(35)
Introduction	158	(1)
Points	159	(6)
Interior Points	159	(2)
Isolated Points	161	(1)
Points of Accumulation	161	(1)
Boundary Points	162	(3)
Sets	165	(8)
Closed Sets	166	(1)
Open Sets	167	(6)
Elementary Topology	173	(3)
Compactness Arguments	176	(13)
Bolzano-Weierstrass Property	178	(1)
†Cantor's Intersection Property	179	(2)
†Cousin's Property	181	(1)
*Heine-Borel Property	182	(4)
Compact Sets	186	(3)
Countable Sets	189	(1)
Challenging Problems for Chapter 4	190	(3)
Continuous Functions	193	(60)
Introduction to Limits	193	(11)
Limits (&epsis;-δ Definition)	193	(4)
Limits (Sequential Definition)	197	(3)
†Limits (Mapping Definition)	200	(1)
One-Sided Limits	201	(2)
Infinite Limits	203	(1)
Properties of Limits	204	(18)
Uniqueness of Limits	205	(1)
Boundedness of Limits	205	(2)
Algebra of Limits	207	(3)
Order Properties	210	(3)
Composition of Functions	213	(2)
Examples	215	(7)
*Limits Superior and Inferior	222	(1)
Continuity	223	(12)
†How to Define Continuity	223	(4)
Continuity at a Point	227	(3)
Continuity at an Arbitrary Point	230	(2)
†Continuity on a Set	232	(3)
Properties of Continuous Functions	235	(1)
Uniform Continuity	236	(4)
Extremal Properties	240	(1)
Darboux Property	241	(2)
Points of Discontinuity	243	(8)
Types of Discontinuity	243	(2)
Monotonic Functions	245	(4)
*How Many Points of Discontinuity?	249	(2)
Challenging Problems for Chapter 5	251	(2)
More on Continuous Functions and Sets	253	(33)
Introduction	253	(1)
Dense Sets	253	(2)
Nowhere Dense Sets	255	(2)
*The Baire Category Theorem	257	(5)
*A Two-Player Game	257	(2)
*The Baire Category Theorem	259	(1)
*Uniform Boundedness	260	(2)
*Cantor Sets	262	(7)
*Construction of the Cantor Ternary Set	262	(3)
†An Arithmetic Construction of K	265	(2)
*The Cantor Function	267	(2)
*Borel Sets	269	(4)
*Sets of Type Gδ	269	(2)
*Sets of Type Fσ	271	(2)
*Oscillation and Continuity	273	(6)
*Oscillation of a Function	274	(3)
*The Set of Continuity Points	277	(2)
*Sets of Measure Zero	279	(6)
Challenging Problems for Chapter 6	285	(1)
Differentiation	286	(60)
Introduction	286	(1)
The Derivative	286	(8)
Definition of the Derivative	287	(5)
Differentiability and Continuity	292	(1)
†The Derivative as a Magnification	293	(1)
Computations of Derivatives	294	(11)
Algebraic Rules	295	(3)
The Chain Rule	298	(3)
Inverse Functions	301	(2)
The Power Rule	303	(2)
Continuity of the Derivative?	305	(2)
Local Extrema	307	(2)
Mean Value Theorem	309	(6)
Rolle's Theorem	310	(2)
Mean Value Theorem	312	(2)
†Cauchy's Mean Value Theorem	314	(1)
Monotonicity	315	(3)
*Dini Derivates	318	(4)
The Darboux Property of the Derivative	322	(3)
Convexity	325	(5)
†L'Hopital's Rule	330	(9)
†L'Hopital's Rule: 0/0 Form	332	(2)
†L'Hopital's Rule as x → ∞	334	(2)
†L'Hopital's Rule: ∞/∞ Form	336	(3)
†Taylor Polynomials	339	(4)
Challenging Problems for Chapter 7	343	(3)
The Integral	346	(38)
Introduction	346	(3)
Cauchy's First Method	349	(5)
†Scope of Cauchy's First Method	351	(3)
Properties of the Integral	354	(5)
Cauchy's Second Method	359	(3)
Cauchy's Second Method (Continued)	362	(2)
The Riemann Integral	364	(10)
Some Examples	366	(2)
*Riemann's Criteria	368	(2)
*Lebesgue's Criterion	370	(3)
*What Functions Are Riemann Integrable?	373	(1)
Properties of the Riemann Integral	374	(4)
†The Improper Riemann Integral	378	(2)
†More on the Fundamental Theorem of Calculus	380	(2)
Challenging Problems for Chapter 8	382	(2)
Sequences and Series of Functions	384	(42)
Introduction	384	(1)
Pointwise Limits	385	(6)
Uniform Limits	391	(13)
The Cauchy Criterion	394	(2)
Weierstrass M-Test	396	(2)
*Abel's Test for Uniform Convergence	398	(6)
Uniform Convergence and Continuity	404	(4)
*Dini's Theorem	405	(3)
Uniform Convergence and the Integral	408	(7)
Sequences of Continuous Functions	408	(2)
†Sequences of Riemann Integrable Functions	410	(2)
Sequences of Improper Integrals	412	(3)
Uniform Convergence and Derivatives	415	(4)
†Limits of Discontinuous Derivatives	417	(2)
†Pompeiu's Function	419	(3)
*Continuity and Pointwise Limits	422	(3)
Challenging Problems for Chapter 9	425	(1)
Power Series	426	(36)
Introduction	426	(1)
Power Series: Convergence	427	(5)
Uniform Convergence	432	(3)
Functions Represented by Power Series	435	(8)
Continuity of Power Series	435	(1)
Integration of Power Series	436	(1)
Differentiation of Power Series	437	(3)
Power Series Representations	440	(3)
The Taylor Series	443	(6)
Representing a Function by a Taylor Series	444	(3)
Analytic Functions	447	(2)
Products of Power Series	449	(3)
Quotients of Power Series	450	(2)
Composition of Power Series	452	(1)
†Trigonometric Series	453	(9)
†Uniform Convergence of Trigonometric Series	454	(1)
†Fourier Series	455	(1)
†Convergence of Fourier Series	456	(4)
†Weierstrass Approximation Theorem	460	(2)
The Euclidean Spaces Rn	462	(33)
The Algebraic Structure of Rn	462	(2)
The Metric Structure of Rn	464	(4)
Elementary Topology of Rn	468	(2)
Sequences in Rn	470	(5)
Functions and Mappings	475	(5)
Functions from Rn → R	475	(2)
Functions from Rn → Rm	477	(3)
Limits of Functions from Rn → Rm	480	(6)
Definition	480	(3)
Coordinate-Wise Convergence	483	(2)
Algebraic Properties	485	(1)
Continuity of Functions from Rn to Rm	486	(3)
Compact Sets in Rn	489	(1)
Continuous Functions on Compact Sets	490	(1)
†Additional Remarks	491	(4)
Differentiation on Rn	495	(78)
Introduction	495	(1)
Partial and Directional Derivatives	496	(10)
Partial Derivatives	497	(3)
Directional Derivatives	500	(1)
Cross Partials	501	(5)
Integrals Depending on a Parameter	506	(4)
Differentiable Functions	510	(16)
Approximation by Linear Functions	511	(1)
Definition of Differentiability	512	(4)
Differentiability and Continuity	516	(1)
Directional Derivatives	517	(2)
An Example	519	(2)
Sufficient Conditions for Differentiability	521	(2)
The Differential	523	(3)
Chain Rules	526	(15)
Preliminary Discussion	526	(4)
Informal Proof of a Chain Rule	530	(1)
Notation of Chain Rules	531	(2)
Proofs of Chain Rules (I)	533	(2)
Mean Value Theorem	535	(1)
Proofs of Chain Rules (II)	536	(2)
Higher Derivatives	538	(3)
Implicit Function Theorems	541	(15)
One-Variable Case	542	(3)
Several-Variable Case	545	(4)
Simultaneous Equations	549	(4)
Inverse Function Theorem	553	(3)
Functions From R → Rm	556	(4)
*Functions From Rn → Rm	560	(13)
†Review of Differentials and Derivatives	560	(3)
†Definition of the Derivative	563	(1)
†Jacobians	564	(3)
†Chain Rules	567	(3)
†Proof of Chain Rule	570	(3)
Metric Spaces	573
Introduction	573	(2)
Metric Spaces---Specific Examples	575	(5)
Additional Examples	580	(5)
Sequence Spaces	580	(2)
Function Spaces	582	(3)
Convergence	585	(4)
Sets in a Metric Space	589	(8)
Functions	597	(16)
Continuity	599	(5)
Homeomorphisms	604	(6)
Isometries	610	(3)
Separable Spaces	613	(3)
Complete Spaces	616	(7)
Completeness Proofs	617	(2)
Subspaces of a Complete Space	619	(1)
†Cantor Intersection Property	619	(1)
†Completion of a Metric Space	620	(3)
Contraction Maps	623	(7)
†Applications of Contraction Maps (I)	630	(3)
†Applications of Contraction Maps (II)	633	(7)
†Systems of Equations (Example 13.79 Revisited)	634	(1)
†Infinite Systems (Example 13.80 revisited)	635	(2)
†Integral Equations (Example 13.81 revisited)	637	(1)
†Picard's Theorem (Example 13.82 revisited)	638	(2)
Compactness	640	(19)
The Bolzano-Weierstrass Property	641	(3)
Continuous Functions on Compact Sets	644	(2)
†The Heine-Borel Property	646	(2)
†Total Boundedness	648	(3)
†Compact Sets in C[a, b]	651	(5)
†Peano's Theorem	656	(3)
Baire Category Theorem	659	(7)
Nowhere Dense Sets	660	(3)
The Baire Category Theorem	663	(3)
†Applications of the Baire Category Theorem	666	(8)
Functions Whose Graphs ``Cross No Lines''	667	(3)
Nowhere Monotonic Functions	670	(1)
Continuous Nowhere Differentiable Functions	671	(1)
Cantor Sets	672	(2)
Challenging Problems for Chapter 13	674
APPENDIX A: BACKGROUND	A-1
A.1 Should I Read This Chapter?	A-1
A.2 Notation	A-1
A.2.1 Set Notation	A-1
A.2.2 Function Notation	A-5
A.3 What Is Analysis?	A-11
A.4 Why Proofs?	A-12
A.5 Indirect Proof	A-13
A.6 Contraposition	A-14
A.7 Counterexamples	A-15
A.8 Induction	A-17
A.9 Quantifiers	A-19
APPENDIX B: HINTS FOR SELECTED EXERCISES	A-22
Subject Index	A-48

Excerpts

PREFACE University mathematics departments have for many years offered courses with titles such asAdvanced CalculusorIntroductory Real Analysis.These courses are taken by a variety of students, serve a number of purposes, and are written at various levels of sophistication. The students range from ones who have just completed a course in elementary calculus to beginning graduate students in mathematics. The purposes are multifold: To present familiar concepts from calculus at a more rigorous level. To introduce concepts that are not studied in elementary calculus but that are needed in more advanced undergraduate courses. This would include such topics as point set theory, uniform continuity of functions, and uniform convergence of sequences of functions. To provide students with a level of mathematical sophistication that will prepare them for graduate work in mathematical analysis, or for graduate work in several applied fields such as engineering or economics. To develop many of the topics that the authors feel all students of mathematics should know. There are now many texts that address some or all of these objectives. These books range from ones that do little more than address objective (1) to ones that try to address all four objectives. The books of the first extreme are generally aimed at one-term courses for students with minimal background. Books at the other extreme often contain substantially more material than can be covered in a one-year course. The level of rigor varies considerably from one book to another, as does the style of presentation. Some books endeavor to give a very efficient streamlined development; others try to be more user friendly. We have opted for the user-friendly approach. We feel this approach makes the concepts more meaningful to the student. Our experience with students at various levels has shown that most students have difficulties when topics that are entirely new to them first appear. For some students that might occur almost immediately when rigorous proofs are required. For others, the difficulties begin with elementary point set theory, compactness arguments, and the like. To help students with the transition from elementary calculus to a more rigorous course, we have included motivation for concepts most students have not seen before and provided more details in proofs when we introduce new methods. In addition, we have tried to give students ample opportunity to see the new tools in action. For example, students often feel uneasy when they first encounter the various compactness arguments (Heine-Borel theorem, Bolzano-Weierstrass theorem, Cousin's lemma, introduced in Section 4.5). To help the student see why such theorems are useful, we pose the problem of determining circumstances under which local boundedness of a functionfon a setEimplies global boundedness offonE.We show by example that some conditions onEare needed, namely thatEbe closed and bounded, and then show how each of several theorems could be used to show that closed and boundedness of the setEsuffices. Thus we introduce students to the theorems by showing how the theorems can be used in natural ways to solve a problem. We have also included some optional material, marked as "Advanced" or "Enrichment" and flagged with a scissors symbol. Enrichment We have indicated as "Enrichment"' some relatively elementary material that could be added to a longer course to provide enrichment and additional examples. For example, in Chapter 3 we have added to the study of series a section on infinite products. While such a topic plays an important role in the representation of analytic functions, it is presented here to allow the instructor to explore ideas that are closely related to the study of series and that help illustrate and review many of the fundamenta