A First Course in Real Analysis

1 The real number system.- 1.1 Axioms for a field.- 1.2 Natural numbers, sequences, and extensions.- 1.3 Inequalities.- 1.4 Mathematical induction-definition of natural number.- 2 Continuity and limits.- 2.1 Continuity.- 2.2 Theorems on limits.- 2.3 One-sided limits-continuity on sets.- 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 Cauchy sequences and the Cauchy criterion.- 3.7 The Heine-Borel and Lebesgue theorems.- 4 Elementary theory of differentiation.- 4.1 Differentiation of functions on ?1.- 4.2 Inverse functions.- 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.- 6 Metric spaces and mappings.- 6.1 The Schwarz and Triangle inequalities-metric spaces.- 6.2 Fundamentals of point set topology.- 6.3 Denumerable sets-countable and uncountable sets.- 6.4 Compact sets and the Heine-Borel theorem.- 6.5 Functions defined on compact sets.- 6.6 Connected sets.- 6.7 Mappings from one metric space to another.- 7 Differentiation in ?N.- 7.1 Partial derivatives.- 7.2 Higher partial derivatives and Taylor's theorem.- 7.3 Differentiation 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 Elementary theorems.- 9.2 Series of positive and negative terms-power series.- 9.3 Uniform convergence.- 9.4 Uniform convergence of series-power series.- 9.5 Unordered sums.- 9.6 The Comparison test for unordered sums-uniform convergence.- 9.7 Multiple sequences and series.- 10 Fourier series.- 10.1 Formal expansions.- 10.2 Fourier sine and cosine series-change of interval.- 10.3 Convergence theorems.- 11 Functions defined by integrals.- 11.1 The derivative of a function defined by an integral.- 11.2 Improper integrals.- 11.3 Functions defined by improper integrals-the Gamma function.- 12 Functions of bounded variation and the Riemann-Stieltjes integral.- 12.1 Functions of bounded variation.- 12.2 The Riemann-Stieltjes integral.- 13 Contraction mappings and differential equations.- 13.1 Fixed point theorem.- 13.2 Application of the fixed point theorem to differential equations.- 14 Implicit function theorems and differentiable maps.- 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.- 15.1 Complete metric spaces.- 15.2 Convex sets and convex functions.- 15.3 Arzela's theorem: extension of continuous functions.- 15.4 Approximations and the Stone-Weierstrass theorem.- 16 Vector field theory. The theorems of Green a...