Real Variables with Basic Metric Space Topology

INTRODUCTIONBasic TerminologyFinite and Infinite Sets; Countably Infinite and Uncountably Infinite SetsDistance and ConvergenceMinicourse in Basic LogicLimit Points and ClosureReview Problems for Chapter 1SOME BASIC TOPOLOGICAL PROPERTIES OF RpUnions and Intersections of Open and Closed SetsCompactnessSome Applications of CompactnessLeast Upper Bounds and CompletenessReview Problems for Chapter 2UPPER AND LOWER LIMITS OF SEQUENCES OF REAL NUMBERSGeneralization of the Limit ConceptSome Properties of Upper and Lower LimitsConvergence of Power SeriesReview Problems for Chapter 3CONTINUOUS FUNCTIONSContinuity: Ideas, Basic Terminology, PropertiesContinuity and CompactnessTypes of DiscontinuitiesThe Cantor SetReview Problems for Chapter 4DIFFERENTIATIONThe Derivative and Its Basic PropertiesAdditional Properties of the Derivative; Some Applications of the Mean Value TheoremReview Problems for Chapter 5RIEMANN-STIELTJES INTEGRATIONDefinition of the IntegralProperties of the IntegralFunctions of Bounded VariationSome Useful Integration TheoremsReview Problems for Chapter 6UNIFORM CONVERGENCE AND APPLICATIONSPointwise and Uniform ConvergenceUniform Convergence and Limit OperationsThe Weierstrass M-test and ApplicationsEquicontinuity and the Arzela-Ascoli TheoremThe Weierstrass Approximation TheoremReview Problems for Chapter 7FURTHER TOPOLOGICAL RESULTSThe Extension ProblemBaire Category TheoremConnectednessSemicontinuous FunctionsReview Problems for Chapter 8EPILOGUESome Compactness ResultsReplacing Cantor's Nested Set PropertyThe Real Numbers RevisitedSOLUTIONS TO PROBLEMSINDEX