A First Course in Real Analysis

This book is designed for a first course in real analysis following the standard course in elementary calculus. The book contains a full treatment (with many illustrative samples and exercises) of the standard topics in infinite metric spaces, and vector field theory. This new edition includes an assortment of new exercises and provides answers for the od-numbered problems.

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Many changes have been made in this second edition of A First Course in Real Analysis. The most noticeable is the addition of many problems and the inclusion of answers to most of the odd-numbered exercises. The book's readability has also been improved by the further clarification of many of the proofs, additional explanatory remarks, and clearer notation.

Contenu
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 BolzanoWeierstrass 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 HeineBorel 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 RiemannStieltjes Integral and Functions of Bounded Variation.- 12.1 Functions of Bounded Variation.- 12.2 The RiemannStieltjes 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 StoneWeierstrass 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.