A First Course in Real Analysis

1 The Real Number System.- 1.1 Axioms for a Field.- 1.2 Natural Numbers and Sequences.- 1.3 Inequalities.- 1.4 Mathematical Induction.- 2 Continuity And Limits.- 2.1 Continuity.- 2.2 Limits.- 2.3 One-Sided Limits.- 2.4 Limits at Infinity; Infinite Limits.- 2.5 Limits of Sequences.- 3 Basic Properties of Functions on ?1.- 3.1 The Intermediate-Value Theorem.- 3.2 Least Upper Bound; Greatest Lower Bound.- 3.3 The Bolzano—Weierstrass Theorem.- 3.4 The Boundedness and Extreme-Value Theorems.- 3.5 Uniform Continuity.- 3.6 The Cauchy Criterion.- 3.7 The Heine-Borel and Lebesgue Theorems.- 4 Elementary Theory of Differentiation.- 4.1 The Derivative in ?1.- 4.2 Inverse Functions in ?1.- 5 Elementary Theory of Integration.- 5.1 The Darboux Integral for Functions on ?1.- 5.2 The Riemann Integral.- 5.3 The Logarithm and Exponential Functions.- 5.4 Jordan Content and Area.- 6 Elementary Theory of Metric Spaces.- 6.1 The Schwarz and Triangle Inequalities; Metric Spaces.- 6.2 Elements of Point Set Topology.- 6.3 Countable and Uncountable Sets.- 6.4 Compact Sets and the Heine—Borel Theorem.- 6.5 Functions on Compact Sets.- 6.6 Connected Sets.- 6.7 Mappings from One Metric Space to Another.- 7 Differentiation in ?N.- 7.1 Partial Derivatives and the Chain Rule.- 7.2 Taylor’s Theorem; Maxima and Minima 178.- 7.3 The Derivative in ?N.- 8 Integration in ?N.- 8.1 Volume in ?N.- 8.2 The Darboux Integral in ?N.- 8.3 The Riemann Integral in ?N.- 9 Infinite Sequences and Infinite Series.- 9.1 Tests for Convergence and Divergence.- 9.2 Series of Positive and Negative Terms; Power Series.- 9.3 Uniform Convergence of Sequences.- 9.4 Uniform Convergence of Series; Power Series.- 9.5 Unordered Sums.- 9.6 The Comparison Test for Unordered Sums; Uniform Convergence.- 9.7Multiple Sequences and Series.- 10 Fourier Series.- 10.1 Expansions of Periodic Functions.- 10.2 Sine Series and Cosine Series; Change of Interval.- 10.3 Convergence Theorems.- 11 Functions Defined by Integrals; Improper Integrals.- 11.1 The Derivative of a Function Defined by an Integral; the Leibniz Rule.- 1l.2 Convergence and Divergence of Improper Integrals.- 11.3 The Derivative of Functions Defined by Improper Integrals; the Gamma Function.- 12 The Riemann—Stieltjes Integral and Functions of Bounded Variation.- 12.1 Functions of Bounded Variation.- 12.2 The Riemann—Stieltjes Integral.- 13 Contraction Mappings, Newton’s Method, and Differential Equations.- 13.1 A Fixed Point Theorem and Newton’s Method.- 13.2 Application of the Fixed Point Theorem to Differential Equations.- 14 Implicit Function Theorems and Lagrange Multipliers.- 14.1 The Implicit Function Theorem for a Single Equation.- 14.2 The Implicit Function Theorem for Systems.- 14.3 Change of Variables in a Multiple Integral.- 14.4 The Lagrange Multiplier Rule.- 15 Functions on Metric Spaces; Approximation.- 15.1 Complete Metric Spaces.- 15.2 Convex Sets and Convex Functions.- 15.3 Arzela’s Theorem; the Tietze Extension Theorem.- 15.4 Approximations and the Stone—Weierstrass Theorem.- 16 Vector Field Theory; the Theorems of Green and Stokes.- 16.1 Vector Functions on ?1.- 16.2 Vector Functions and Fields on ?N.- 16.3 Line Integrals in ?N.- 16.4 Green’s Theorem in the Plane.- 16.5 Surfaces in ?3; Parametric Representation.- 16.6 Area of a Surface in ?3; Surface Integrals.- 16.7 Orientable Surfaces.- 16.8 The Stokes Theorem.- 16.9 The Divergence Theorem.- Appendixes.- Appendix 1 Absolute Value.- Appendix 2 Solution of Algebraic Inequalities.- Appendix 3 Expansions of Real Numbers inAny Base.- Answers to Odd-Numbered Problems.