A Basic Course in Real Analysis


Lecture 1 - Rational Numbers and Rational Cuts


Lecture 2 - Irrational numbers, Dedekind's Theorem


Lecture 3 - Continuum and Exercises


Lecture 4 - Continuum and Exercises (Continued.)


Lecture 5 - Cantor's Theory of Irrational Numbers


Lecture 6 - Cantor's Theory of Irrational Numbers (Continued.)


Lecture 7 - Equivalence of Dedekind and Cantor's Theory


Lecture 8 - Finite, Infinite, Countable and Uncountable Sets of Real Numbers


Lecture 9 - Types of Sets with Examples, Metric Space


Lecture 10 - Various properties of open set, closure of a set


Lecture 11 - Ordered set, Least upper bound, greatest lower bound of a set


Lecture 12 - Compact Sets and its properties


Lecture 13 - Weiersstrass Theorem, Heine Borel Theorem, Connected set


Lecture 14 - Tutorial - II


Lecture 15 - Concept of limit of a sequence


Lecture 16 - Some Important limits, Ratio tests for sequences of Real Numbers


Lecture 17 - Cauchy theorems on limit of sequences with examples


Lecture 18 - Fundamental theorems on limits, Bolzano-Weiersstrass Theorem


Lecture 19 - Theorems on Convergent and divergent sequences


Lecture 20 - Cauchy sequence and its properties


Lecture 21 - Infinite series of real numbers


Lecture 22 - Comparison tests for series, Absolutely convergent and Conditional convergent series


Lecture 23 - Tests for absolutely convergent series


Lecture 24 - Raabe's test, limit of functions, Cluster point


Lecture 25 - Some results on limit of functions


Lecture 26 - Limit Theorems for functions


Lecture 27 - Extension of limit concept (one sided limits)


Lecture 28 - Continuity of Functions


Lecture 29 - Properties of Continuous Functions


Lecture 30 - Boundedness Theorem, Max-Min Theorem and Bolzano's theorem


Lecture 31 - Uniform Continuity and Absolute Continuity


Lecture 32 - Types of Discontinuities, Continuity and Compactness


Lecture 33 - Continuity and Compactness (Continued.), Connectedness


Lecture 34 - Differentiability of real valued function, Mean Value Theorem


Lecture 35 - Mean Value Theorem (Continued.)


Lecture 36 - Application of MVT , Darboux Theorem, L Hospital Rule


Lecture 37 - L'Hospital Rule and Taylor's Theorem


Lecture 38 - Tutorial - III


Lecture 39 - Riemann/Riemann Stieltjes Integral


Lecture 40 - Existence of Reimann Stieltjes Integral


Lecture 41 - Properties of Reimann Stieltjes Integral


Lecture 42 - Properties of Reimann Stieltjes Integral (Continued.)


Lecture 43 - Definite and Indefinite Integral


Lecture 44 - Fundamental Theorems of Integral Calculus


Lecture 45 - Improper Integrals


Lecture 46 - Convergence Test for Improper Integrals