Elements of real analysis

PREFACEPRELIMINARIES Sets FunctionsREAL NUMBERSField AxiomsOrder Axioms Natural Numbers, Integers, Rational NumbersCompleteness Axiom Decimal Representation of Real Numbers Countable SetsSEQUENCESSequences and ConvergenceProperties of Convergent Sequences Monotonic SequencesThe Cauchy Criterion Subsequences Upper and Lower Limits Open and Closed Sets INFINITE SERIESBasic Properties Convergence TestsLIMIT OF A FUNCTIONLimit of a Function Basic Theorems Some Extensions of the LimitMonotonic Functions CONTINUITY Continuous Functions Combinations of Continuous Functions Continuity on an IntervalUniformContinuityCompact Sets and ContinuityDIFFERENTIATION The DerivativeTheMean Value TheoremL'Hôpital's RuleTaylor's TheoremTHE RIEMANN INTEGRALRiemann Integrability Darboux's Theorem and Riemann SumsProperties of the Integral The Fundamental Theorem of Calculus Improper IntegralsSEQUENCES AND SERIES OF FUNCTIONSSequences of FunctionsProperties of Uniform ConvergenceSeries of FunctionsPower Series LEBESGUE MEASURE Classes of Subsets of R Lebesgue Outer Measure Lebesgue Measure Measurable Functions LEBESGUE INTEGRATIONDefinition of the Lebesgue IntegralProperties of the Lebesgue Integral Lebesgue Integral and Pointwise ConvergenceLebesgue and Riemann IntegralsREFERENCESNOTATIONINDEX.