This is a component considered one of a two-volume ebook on genuine research and is meant for senior undergraduate scholars of arithmetic who've already been uncovered to calculus. The emphasis is on rigour and foundations of study. starting with the development of the quantity structures and set thought, the ebook discusses the fundamentals of study (limits, sequence, continuity, differentiation, Riemann integration), via to strength sequence, numerous variable calculus and Fourier research, after which ultimately the Lebesgue fundamental. those are nearly totally set within the concrete atmosphere of the genuine line and Euclidean areas, even though there's a few fabric on summary metric and topological areas. The ebook additionally has appendices on mathematical common sense and the decimal process. the total textual content (omitting a few much less primary issues) will be taught in quarters of 25–30 lectures each one. The direction fabric is deeply intertwined with the workouts, because it is meant that the scholar actively examine the fabric (and perform considering and writing carefully) by way of proving a number of of the major leads to the theory.
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