A Basic Course in Real Analysis

Srivastava, P. D.

Modules/Lectures: Rational numbers & Rational cuts; Irrational numbers, Dedekind's Theorem; Continuum & Exercises; Cantor’s Theory of Irrational numbers; Equivalence of Dedekind & Cantor's Theory; finite, infinite, countable and uncountable sets of real numbers; Types of sets with examples, Metric space; Various properties of open set, closure of a set; Ordered set, least upper bound, greatest lower bound of set; compact sets and its properties; Weierstrass Theorem, Heine Borel Theorem, connected set; Tutorial II; Concept of limit of a sequence; Some important limits, ratio tests for sequences of real numbers; Cauchy theorem on limit of sequences with examples; fundamental theorems on limit, Bolzano-Weiersstrass theorem; theorems on convergent and divergent sequences; Cauchy sequence & its properties; Infinite series of real numbers; comparison test for series, Absolutely convergent and conditional convergent series; tests for absolutely convergent series; Raabe's test, limit of functions, cluster point; some results on limit of functions; limit theorems for functions; extension of limit concept(one sided li mits); continuity of functions; properties of continuous functions; boundedness theorem, max-min theorem and Bolzano's theorem; uniform continuity and absolute continuity; types of discontinuities, continuity and compactness; continuity and compactness, connectedness; differentiability of real valued function, mean value theorem; application of MVT, Darboux theorem, L Hospital Rule; L'Hospital Rule and Taylor's theorem; tutorial III; Riemann/Riemann Stieltjes Integral; Existence and Properties of Riemann Stieltjes Integral; Definite and Indefinite Integral; Fundamental theorem of Integral calculus; Improper Integrals; Convergence test for Improper Integrals.

Published by:

National Programme on Technology Enhanced Learning (NPTEL)

DOER Persistent Identifier: http://doer.col.org/handle//123456789/3561