Thomson B.S., Bruckner J.B., Bruckner A.M. Elementary Real Analysis

CreateSpace, 2008. — 684 p. — ISBN10: 143484367X; ISBN13: 978-1434843678This version of Elementary Real Analysis, Second Edition, is a hypertexted PDF file, suitable for on-screen viewing.This book is the second edition of an undergraduate level Real Analysis textbook formerly published by Prentice Hall (Pearson) in 2001. It is designed to be a user-friendly text that is suitable for a one year course.
Additional elementary material designated as enrichment can be included for students with minimal background. Material designated as advanced is intended for students with stronger backgrounds. Such topics are of a more sophisticated nature.Volume one
Properties of the real numbersThe Real Number System
Algebraic Structure
Order Structure
Bounds
Sups and Infs
The Archimedean Property
Inductive Property of IN
The Rational Numbers Are Dense
The Metric Structure of R
Challenging Problems for Chapter
NotesSequencesSequences
Sequence Examples
Countable Sets
Convergence
Divergence
Boundedness Properties of Limits
Algebra of Limits
Order Properties of Limits
Monotone Convergence Criterion
Examples of Limits
Subsequences
Cauchy Convergence Criterion
Upper and Lower Limits
Challenging Problems for Chapter
NotesInfinite sumsFinite Sums
Infinite Unordered sums
Cauchy Criterion
Ordered Sums: Series
Properties
Special Series
Criteria for Convergence
Boundedness Criterion
Cauchy Criterion
Absolute Convergence
Tests for Convergence
Trivial Test
Direct Comparison Tests
Limit Comparison Tests
Ratio Comparison Test
d’Alembert’s Ratio Test
Cauchy’s Root Test
Cauchy’s Condensation Test
Integral Test
Kummer’s Tests
Raabe’s Ratio Test
Gauss’s Ratio Test
Alternating Series Test
Dirichlet’s Test
Abel’s Test
Rearrangements
Unconditional Convergence
Conditional Convergence
Comparison of ∑i=∞ai
Products of Series
Products of Absolutely Convergent Series
Products of Nonabsolutely Convergent Series
Summability Methods
Ces`aro’s Method
Abel’s Method
More on Infinite Sums
Infinite Products
Challenging Problems for Chapter
NotesSets of real numbersPoints
Interior Points
Isolated Points
Points of Accumulation
Boundary Points
Sets
Closed Sets
Open Sets
Elementary Topology
Compactness Arguments
Bolzano-Weierstrass Property
Cantor’s Intersection Property
Cousin’s Property
Heine-Borel Property
Compact Sets
Countable Sets
Challenging Problems for Chapter
NotesContinuous functions
Introduction to Limits
Limits (ε-δ Definition)
Limits (Sequential Definition)
Limits (Mapping Definition)
One-Sided Limits
Infinite Limits
Properties of Limits
Uniqueness of Limits
Boundedness of Limits
Algebra of Limits
Order Properties
Composition of Functions
Examples
Limits Superior and Inferior
Continuity
How to Define Continuity
Continuity at a Point
Continuity at an Arbitrary Point
Continuity on a Set
Properties of Continuous Functions
Uniform Continuity
Extremal Properties
Darboux Property
Points of Discontinuity
Types of Discontinuity
Monotonic Functions
How Many Points of Discontinuity?
Challenging Problems for Chapter
NotesMore on continuous functions and setsDense Sets
Nowhere Dense Sets
The Baire Category Theorem
A Two-Player Game
The Baire Category Theorem
Uniform Boundedness
Cantor Sets
Construction of the Cantor Ternary Set
An Arithmetic Construction of K
The Cantor Function
Borel Sets
Sets of Type G_
Sets of Type F_
Oscillation and Continuity
Oscillation of a Function
The Set of Continuity Points
Sets of Measure Zero
Challenging Problems for Chapter
NotesDifferentiationThe Derivative
Definition of the Derivative
Differentiability and Continuity
The Derivative as a Magnification
Computations of Derivatives
Algebraic Rules
The Chain Rule
Inverse Functions
The Power Rule
Continuity of the Derivative?
Local Extrema
Mean Value Theorem
Rolle’s Theorem
Mean Value Theorem
Cauchy’s Mean Value Theorem
Monotonicity
Dini Derivates
The Darboux Property of the Derivative
Convexity
L’Hopital’s Rule
L’Hopital’s Rule:
L’Hˆopital’s
L’Hopital’s Rule
Taylor Polynomials
Challenging Problems for Chapter
NotesThe integralCauchy’s First Method
Scope of Cauchy’s First Method
Properties of the Integral
Cauchy’s Second Method
Cauchy’s Second Method (Continued)
The Riemann Integral
Some Examples
Riemann’s Criteria
Lebesgue’s Criterion
What Functions Are Riemann Integrable?
Properties of the Riemann Integral
The Improper Riemann Integral
More on the Fundamental Theorem of Calculus
Challenging Problems for Chapter
NotesVolume twoSequences and series of functionsPointwise Limits
Uniform Limits
The Cauchy Criterion
Weierstrass M-Test
Abel’s Test for Uniform Convergence
Uniform Convergence and Continuity
Dini’s Theorem
Uniform Convergence and the Integral
Sequences of Continuous Functions
Sequences of Riemann Integrable Functions
Sequences of Improper Integrals
Uniform Convergence and Derivatives
Limits of Discontinuous Derivatives
Pompeiu’s Function
Continuity and Pointwise Limits
Challenging Problems for Chapter
NotesPower seriesPower Series: Convergence
Uniform Convergence
Functions Represented by Power Series
Continuity of Power Series
Integration of Power Series
Differentiation of Power Series
Power Series Representations
The Taylor Series
Representing a Function by a Taylor Series
Analytic Functions
Products of Power Series
Quotients of Power Series
Composition of Power Series
Trigonometric Series
Uniform Convergence of Trigonometric Series
Fourier Series
Convergence of Fourier Series
Weierstrass Approximation Theorem
NotesThe euclidean spaces RN
The Algebraic Structure of Rn
The Metric Structure of Rn
Elementary Topology of Rn
Sequences in Rn
Functions and Mappings
Functions from Rn → R
Functions from Rn → Rm
Limits of Functions from Rn → Rm
Definition
Coordinate-Wise Convergence
Algebraic Properties
Continuity of Functions from Rn to Rm
Compact Sets in Rn
Continuous Functions on Compact Sets
Additional Remarks
NotesDifferentiation on RNPartial and Directional Derivatives
Partial Derivatives
Directional Derivatives
Cross Partials
Integrals Depending on a Parameter
Differentiable Functions
Approximation by Linear Functions
Definition of Differentiability
Differentiability and Continuity
Directional Derivatives
An Example
Sufficient Conditions for Differentiability
The Differential
Chain Rules
Preliminary Discussion
Informal Proof of a Chain Rule
Notation of Chain Rules
Proofs of Chain Rules (I)
Mean Value Theorem
Proofs of Chain Rules (II)
Higher Derivatives
Implicit Function Theorems
One-Variable Case
Several-Variable Case
Simultaneous Equations
Inverse Function Theorem
Functions From R → Rm
Functions From Rn → Rm
Review of Differentials and Derivatives
Definition of the Derivative
Jacobians
Chain Rules
Proof of Chain Rule
NotesMetric spacesMetric Spaces—Specific Examples
Additional Examples
Sequence Spaces
Function Spaces
Convergence
Sets in a Metric Space
Functions
Continuity
Homeomorphisms
Isometries
Separable Spaces
Complete Spaces
Completeness Proofs
Subspaces of a Complete Space
Cantor Intersection Property
Completion of a Metric Space
Contraction Maps
Applications of Contraction Maps (I)
Applications of Contraction Maps (II)
Systems of Equations (Example Revisited)
Infinite Systems (Example revisited)
Integral Equations (Example revisited)
Picard’s Theorem (Example revisited)
Compactness
The Bolzano-Weierstrass Property
Continuous Functions on Compact Sets
The Heine-Borel Property
Total Boundedness
Compact Sets in C[a, b]
Peano’s Theorem
Baire Category Theorem
Nowhere Dense Sets
The Baire Category Theorem
Applications of the Baire Category Theorem
Functions Whose Graphs “Cross No Lines”
Nowhere Monotonic Functions
Continuous Nowhere Differentiable Functions
Cantor Sets
Challenging Problems for Chapter
NotesVolume one
Appendix: background
Should I Read This Chapter?
Notation
Set Notation
Function Notation
What Is Analysis?
Why Proofs?
Indirect Proof
Contraposition
Counterexamples
Induction
Quantifiers
Notes