A Basic Course in Real Analysis

Overview

Contents:
Dedekind Theory of Irrational numbers : Rational numbers, section of Rational numbers, Irrational numbers, real Numbers, Dedekind Theorem, The Continuum Exercise- Tutorial
Cantors Theory of Irrational numbers : Cantors Theory, Convergent sequence of real numbers, Equivalence of the definition of Dedekind & Cantor
Sets of Points : The upper & lower bounds, l.u.b. & g.l.b. of sets, limiting point, Weierstrass Theorem, Derived sets, Countable & Non constable sets, Cardinal numbers, Open & Closed sets, Closure of a set, Perfect set, Heine-Borel Theorem

Limit of Sequences of Real Numbers : Bounded sequences, Null sequences, Monotone sequences, Convergent sequences, Fundamental theorems on limit, limit sup, limit inf of sequences, Ratio Test & other Tests, Cauchy theorems, Cauchy Convergence Criteria
Exercises- Tutorial
Infinite Series of Real numbers : Introduction of infinite series, Tests for its convergence, Absolute convergence, Conditional convergence
Limit of functions : Concepts of Limit of functions, Limit Theorems, Some extension of Limit Concepts, Exercises- Tutorials

Continuity of Functions : Cauchys and Heines definitions of continuity, Properties of Continuous functions, Uniform continuity, Absolute continuity, Discontinuous Functions, Types of Discontinuities
Differentiability : Concept of Derivatives, Rolles theorem, Mean value theorem, L Hospital Rule, Taylors Theorem Exercises- Tutorial
Riemann Integration / Reimann- Stieltjes Intergral : The Upper and lower R-integrals, Integrable ( R ) functions, Properties of definite and indefinite integral, Mean value theorems, Absolute convergence, convergence, Test for improper integrals. Definition & Existence of the Reimann- Stieltjes Integral & its properties Exercise, Tutorial