Syllabus covered:

Unit 1 [6 hours]

Peano Axioms and Natural Numbers, Axiom of Choice and Set Theory, Russel’s Paradox, Cardinality of Sets, Integers, Rationals, Real Numbers, Functions of Real Numbers.

Unit 2 [16.5 hours]

Sequences of Rational and Real Numbers, Convergence of Sequences, Limits of Sequences, Limit Laws, Suprema and Infima of Sequences, Limit Superior, Limit Inferior and Limit Points of Sequences, Some Convergence Tests of Sequences, Cauchy Sequences, Bounded and Monotone Sequences, Sub sequences and Bolzano-Weierstrass Theorem.

Finite and Infinite Series, Convergence and Absolute Convergence of Infinite Series, Series Laws, Rearrangements of Infinite Series, Some tests of Convergence of Infinite Series: Root Test, Ratio Test, Raabe’s Test, etc.

Countable and Uncountable Sets, Summation on Countable Sets and Fubini’s Theorem, Cantor’s Theorem, Ordered Sets and Zorn’s Lemma, Closure of Sets, Limit Points of a Set, Bounded and Unbounded Sets, Heine-Borel Theorem.

Unit 3 [16.5 hours]

Continuity of Function of Real Numbers, Maxima, Minima, and Maximum Principle for Functions of Real Numbers, Intermediate Value Theorem, Monotonic and Uniformly Continuous Functions, Limits at Infinity.

Differentiability of Functions, Differential Calculus, Local Maxima and Minima of Functions, Mean Value Theorems (Rolle’s Theorem, Cauchy Mean Value Theorem, Lagrange’s Mean Value Theorem), Indeterminate Forms and L'Hôpitals' Rule.

Partition of Sets, Upper and Lower Rieman Integrals, Riemann Sums, Some Integrable and Riemann non Integrable Functions, Two Fundamental Theorems of Calculus.