Week 1: countable & uncountable sets (3 lectures);Concepts of Metric Space (1 lectures);Open ball, closed ball, limit point of a set (1 lectures)
Week 2: Some theorems on Open & closed set (1 lectures);Ordered set, least upper bound, greatest lower bound (2 lectures);Compact set & some properties of Compact set (2 lectures)
Week 3: Heine Borel Theorem (1 lecture);Weierstrass Theorem, connected set (1 lecture);Cantor Set & its properties (1 lecture);Dense set & derived set (1 lecture);Limit of sequences of real numbers & Monotone sequence (1 lecture)
Week 4: Some important limits of sequences (1 lecture);Ratio tests, Cauchy theorems on limits of sequence of real numbers (1 lectures);Fundamental theorems on limit (1 lecture);Some results on limit & Bolzano-Weierstrass Theorem (1 lecture);Criteria for convergent sequence (1 lecture)
Week 5: Criteria for Divergent sequence (1 lecture);Cauchy sequence (1 lecture);Cauchy convergence criteria for sequences (1 lecture);Infinite series of Real numbers (1 lecture);;Convergence Criteria for series of positive real no. (1 lecture)
Week 6: Comparison test for series (1 lecture);Absolutely and Conditional convergent series and Tests (2 lectures);Ratio & Integral Tests for convergence of series (1 lecture);Raabe’s test for convergence of series (1 lecture)
Week 7: Limit of functions & cluster point (2 lectures;Divergence criteria for limit (1 lecture);Various properties of limit of functions (1 lecture);Left & Right hand limits for functions (1 lecture)
Week 8: Limit of functions at infinity (1 lecture);Continuity functions (Cauchy‘s definition) (1 lecture);Continuity functions (Heine‘s definition) (1 lecture);Properties of continuous functions (2 lectures)
Week 9: Boundedness Theorem and Max-Min theorem (1 lecture);Location of root and Bolzano’s theorem (1 lecture);Uniform continuity & related theorems (1 lecture);Absolute continuity& related theorems (1 lecture);Types of discontinuities & Continuity in a Metric Space (1 lectures);
Week 10: Types of discontinuities & Continuity in a Metric Space (1 lectures);Relation between continuity & compact sets (1 lecture);Differentiability of real valued functions (1 lecture);Local Max. – Min. Cauchy’s and Lagrange’s Mean value theorem (1 lecture);Rolle’s Mean value theorems & Applications (1 lecture)
Week 11: Applications of Derivatives (1 lecture);Application of MVT & Darboux’s theorem (1 lecture);L’Hospital Rule (1 lecture);Taylor’s Theorem (1 lecture);Riemann/Riemann Steiltjes Integral (1lecture)
Week 12: Riemann/Riemann Steiltjes Integral (1lecture);Existence of Riemann Stieltjes Integral (1 lecture);Riemann Stieltjes Integrable functions (1 lecture);Properties of Riemann Stieltjes Integral (1 lecture);Various results of Riemann Stieltjes Integral using step function (1 lecture);Some more Results on Riemann Stieltjes Integral (1 lecture)
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