Real Analysis

The course on Real Analysis is designed to provide the learners an understanding of the analytical aspects of various mathematical objects pertaining to the set of real numbers. It covers the basic characterizing properties of real numbers, the sequence and series of real numbers and their convergence and finally, the sequence of real valued functions and their convergence. For pursuing any course on Pure and applied Mathematics or from other disciplines that involve analytical aspects of the Mathematical objects, this course is a must.

Unit 1: Finite and infinite sets, examples of countable and uncountable sets. Real line, field properties of R, bounded sets, suprema and infima, completeness property of R, Archimedean property of R, intervals. Concept of cluster points and statement of Bolzano-Weierstrass theorem. Unit 2: Real Sequence, Bounded sequence, Cauchy convergence criterion for sequences. Cauchy’s theorem on limits, order preservation and squeeze theorem, monotone sequences and their convergence, monotone convergence. Unit 3: Infinite series. Cauchy convergence criterion for series, positive term series, geometric series, comparison test, convergence of p-series, Root test, Ratio test, alternating series, Leibnitz’s test. Definition and examples of absolute and conditional convergence. Unit 4: Sequences and series of functions, Pointwise and uniform convergence. Mn-test, M-test, Statements of the results about uniform convergence and integrability and differentiability of functions, Power series and radius of convergence.