Much of mathematics relies on our ability to be able to solve equations, if not in explicit exact forms, then at least in being able to establish the existence of solutions. To do this requires a knowledge of so-called "analysis", which in many respects is just Calculus in very general settings. Real Analysis-II is a course that develops this basic material in a systematic and rigorous manner in the context of real-valued functions of a real variable. Topics covered are: Real numbers and their basic properties, Sequences: convergence, subsequences, Cauchy sequences, Open, closed, and compact sets of real numbers, Continuous functions and uniform continuity. Lebesgue out measure, Lebesgue integral, sigma algebra of Lebesgue measurable sets.
Students are introduced to fundamental properties of real numbers that lead to the formal development of real analysis.
Students are able to construct rigorous mathematical proofs of basic results in real analysis
Students demonstrate an understanding of limits and how they are used in sequences, series, differentiation and integration
Students are introduced to measure theory and integration: measure space, measurable sets, sigma algebra and Lebegue measure.