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 fourth edition incorporates a large number of additional corrections reported since the release of the third edition, as well as some additional exercises. This is part one of a two-volume introduction to real analysis and is intended for honours undergraduates who have already been exposed to calculus. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory. Later, you can move on with advanced textbooks or follow along. A solid foundation is important for real analysis and I find Terence Tao's books are the best for that. During the courses, I had referred Rudin, Apostol, Bartle and Sherbet etc. The entire text (omitting some less central topics) can be taught in two quarters of twenty-five to thirty lectures each. I have studied real analysis (both introductory and advanced). There are also appendices on mathematical logic and the decimal system. The material starts at the very beginning-the construction of the number systems and set theory-then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and finally to the Lebesgue integral. The emphasis is on rigour and on foundations. This is part one of a two-volume introduction to real analysis and is intended for honours undergraduates who have already been exposed to calculus.
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