Analysis I (Volume 1)

Main subject categories: • Analysis • Numbers • Set Theory • Sequences • Limits of Sequences • Series • Infinite Sets • Continuous Functions • Differentiation • The Riemann IntegralThis two-volume introduction to real analysis is intended for honours undergraduates, who have already been exposed to calculus. The emphasis is on rigour and on foundations. The course material is deeply intertwined with the exercises, as it is intended for the student to actively learn the material and to practice thinking and writing rigorously. This is part one of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system.