Methods of real analysis

PREFACE
CONTENTS
INTRODUCTION
1 SETS AND FUNCTIONS
	1.1 SETS AND ELEMENTS
	1.2 OPERATIONS ON SETS
	1.3 FUNCTIONS
	1.4 REAL-VALUED FUNCTIONS
	1.5 EQUIVALENCE; COUNTABILITY
	1.6 REAL NUMBERS
	1.7 LEAST UPPER BOUNDS
2 SEQUENCES OF REAL NUMBERS
	2.1 DEFINITION OF SEQUENCE AND SUBSEQUENCE
	2.2 LIMIT OF A SEQUENCE
	2.3 CONVERGENT SEQUENCES
	2.4 DIVERGENT SEQUENCES
	2.5 BOUNDED SEQUENCES
	2.6 MONOTONE SEQUENCES
	2.7 OPERATIONS ON CONVERGENT SEQUENCES
	2.8 OPERATIONS ON DIVERGENT SEQUENCES
	2.9 LIMIT SUPERIOR AND LIMIT INFERIOR
	2.10 CAUCHY SEQUENCES
	2.11 SUMMABILITY OF SEQUENCES
	2.12 LIMIT SUPERIOR AND LIMIT INFERIOR FOR SEQUENCES OF SETS
3 SERIES OF REAL NUMBERS
	3.1 CONVERGENCE AND DIVERGENCE
	3.2 SERIES WITH NONNEGATIVE TERMS
	3.3 ALTERNATING SERIES
	3.4 CONDITIONAL CONVERGENCE AND ABSOLUTE CONVERGENCE
	3.5 REARRANGEMENTS OF SERIES
	3.6 TESTS FOR ABSOLUTE CONVERGENCE
	3.7 SERIES WHOSE TERMS FORM A NONINCREASING SEQUENCE
	3.8 SUMMATION BY PARTS
	3.9 (C, 1) SUMMABILITY OF SERIES
	3.10 THE CLASS L^2
	3.11 REAL NUMBERS AND DECIMAL EXPANSIONS
	3.12 NOTES AND ADDITIONAL EXERCISES FOR CHAPTERS 1, 2, AND 3
4 LIMITS AND METRIC SPACES
	4.1 LIMIT OF A FUNCTION ON THE REAL LINE
	4.2 METRIC SPACES
	4.3 LIMITS IN METRIC SPACES
5 CONTINUOUS FUNCTIONS ON METRIC SPACES
	5.1 FUNCTIONS CONTINUOUS AT A POINT ON THE REAL LINE
	5.2 REFORMULATION
	5.3 FUNCTIONS CONTINUOUS ON A METRIC SPACE
	5.4 OPEN SETS
	5.5 CLOSED SETS
	5.6 DISCONTINUOUS FUNCTIONS ON R^1
	5.7 THE DISTANCE FROM A POINT TO A SET
6 CONNECTEDNESS, COMPLETENESS, AND COMPACTNESS
	6.1 MORE ABOUT OPEN SETS
	6.2 CONNECTED SETS
	6.3 BOUNDED SETS AND TOTALLY BOUNDED SETS
	6.4 COMPLETE METRIC SPACES
	6.5 COMPACT METRIC SPACES
	6.6 CONTINUOUS FUNCTIONS ON COMPACT METRIC SPACES
	6.7 CONTINUITY OF THE INVERSE FUNCTION
	6.8 UNIFORM CONTINUITY
	6.9 NOTES AND ADDITIONAL EXERCISES FOR CHAPTERS 4, 5, AND 6
7 CALCULUS
	7.1 SETS OF MEASURE ZERO
	7.2 DEFINITION OF THE RIEMANN INTEGRAL
	7.3 EXISTENCE OF THE RIEMANN INTEGRAL
	7.4 PROPERTIES OF THE RIEMANN INTEGRAL
	7.5 DERIVATIVES
	7.6 ROLLE'S THEOREM
	7.7 THE LAW OF THE MEAN
	7.8 FUNDAMENTAL THEOREMS OF CALCULUS
	7.9IMPROPER INTEGRALS
	7.10 IMPROPER INTEGRALS (CONTINUED)
8 THE ELEMENTARY FUNCTIONS. TAYLOR SERIES
	8.1 HYPERBOLIC FUNCTIONS
	8.2 THE EXPONENTIAL FUNCTION
	8.3 THE LOGARITHMIC FUNCTION. DEFINITION OF x^a
	8.4 THE TRIGONOMETRIC FUNCTIONS
	8.5 TAYLOR'S THEOREM
	8.6THE BINOMIAL THEOREM
	8.7 L'HOSPITAL'S RULE
9 SEQUENCES AND SERIES O F FUNCTIONS
	9.1 POINTWISE CONVERGENCE OF SEQUENCES OF FUNCTIONS
	9.2 UNIFORM CONVERGENCE OF SEQUENCES OF FUNCTIONS
	9.3 CONSEQUENCES OF UNIFORM CONVERGENCE
	9.4 CONVERGENCE AND UNIFORM CONVERGENCE OF SERIES OF FUNCTIONS
	9.5 INTEGRATION AND DIFFERENTIATION OF SERIES OF FUNCTIONS
	9.6 ABEL SUMMABILITY
	9.7 A CONTINUOUS, NOWHERE-DIFFERENTIABLE FUNCTION
10 THREE FAMOUS THEOREMS
	10.1 THE METRIC SPACE C[a, b]
	10.2 THE WEIERSTRASS APPROXIMATION THEOREM
	10.3 PICARD EXISTENCE THEOREM FOR DIFFERENTIAL EQUATIONS
	10.4 THE ARZELA THEOREM ON EQUICONTINUOUS FAMILIES
	10.5 NOTES AND ADDITIONAL EXERCISES FOR CHAPTERS 9 AND 10
11 THE LEBESGUE INTEGRAL
	11.1 LENGTH OF OPEN SETS AND CLOSED SETS
	11.2 INNER AND OUTER MEASURE. MEASURABLE SET
	11.3 PROPERTIES OF MEASURABLE SETS
	11.4 MEASURABLE FUNCTIONS
	11.5 DEFINITION AND EXISTENCE OF THE LEBESGUE INTEGRAL FOR BOUNDED FUNCTIONS
	11.6 PROPERTIES OF THE LEBESGUE INTEGRAL FOR BOUNDED MEASURABLE FUNCTIONS
	11.7 THE LEBESGUE INTEGRAL FOR UNBOUNDED FUNCTIONS
	11.8 SOME FUNDAMENTAL THEOREMS
	11.9 THE METRIC SPACE L^2[a, b]
	11.10 THE INTEGRAL ON -(∞,∞) AND IN THE PLANE
12 FOURIER SERIES
	12.1 DEFINITION OF FOURIER SERIES
	12.2 FORMULATION OF CONVERGENCE PROBLEMS
	12.3 (C, 1) SUMMABILITY OF FOURIER SERIES
	12.4 THE L^2 THEORY OF FOURIER SERIES
	12.5 CONVERGENGE OF FOURIER SERIES
	12.6 ORTHONORMAL EXPANSIONS IN L^2[a, b]
	12.7 NOTES AND ADDITIONAL EXERCISES ON CHAPTERS 11 AND 12
APPENDIX
	ALGEBRA
	ORDER
	THE INTEGERS AND THE RATIONAL NUMBERS
	COMPLETENESS
	ABSOLUTE VALUES
SPECIAL SYMBOLS
INDEX