Larson Lee. Introduction to Real Analysis

University of Louisville, 2014. — 184 p.Updated 13 July 2016About This DocumentBasic Ideas
Sets
Algebra of Sets
Indexed Sets
Functions and Relations
Cardinality
ExercisesThe Real Numbers
The Field Axioms
The Order Axiom
The Completeness Axiom
Comparisons of Q and R
ExercisesSequences
Basic Properties
Monotone Sequences
Subsequences and the Bolzano Weierstrass Theorem
Lower and Upper Limits of a Sequence
The Nested Interval Theorem
Cauchy Sequences
ExercisesSeries
What is a Series?
Positive Series
Absolute and Conditional Convergence
Rearrangements of Series
ExercisesThe Topology of R
Open and Closed Sets
Relative Topologies and Connectedness
Covering Properties and Compactness on R
More Small Sets
ExercisesLimits of Functions
Basic Definitions
Unilateral Limits
Continuity
Unilateral Continuity
Continuous Functions
Uniform Continuity
ExercisesDifferentiation
The Derivative at a Point
Differentiation Rules
Derivatives and Extreme Points
Differentiable Functions
Applications of the Mean Value Theorem
ExercisesIntegration
Partitions
Riemann Sums
Darboux Integration
The Integral
The Cauchy Criterion
Properties of the Integral
The Fundamental Theorem of Calculus
Change of Variables
Integral Mean Value Theorems
ExercisesSequences of Functions
Pointwise Convergence
Uniform Convergence
Metric Properties of Uniform Convergence
Series of Functions
Continuity and Uniform Convergence
Integration and Uniform Convergence
Differentiation and Uniform Convergence
Power Series
ExercisesFourier Series
Trigonometric Polynomials
The Riemann Lebesgue Lemma
The Dirichlet Kernel
Dini’s Test for Point wise Convergence
Gibbs Phenomenon
ADivergent Fourier Series
The Fejér Kernel
ExercisesAppendix Bibliography
Appendix Index