Syllabus covered:

Unit 1 [7.5 hours]

Peano axioms, Addition of natural numbers, Cancellation law, Multiplication of natural numbers, Principle of well ordering, Zermelo-Fraenkel set theory, Russel's paradox, Functions and their composition, Images, Inverse images, Power set, Cardinality, Finite/Infinite sets, Integers, Negation of integers, Rationals, Reciprocal of rationals, Absolute value, Distance between rationals, Division algorithm, epsilon-closeness, Gaps in rational numbers

Unit 2 [7.5 hours]

Division algorithm for integers, Principle of infinite descent, Interspersing of integers by rationals, Interspersing of rationals by rationals (density property of rationals), Gaps in rationals, Sequences of rationals, Cauchy sequences, Equivalence of sequences, Bounded sequences, Real numbers as formal limit of rationals, Addition, multiplication and reciprocal of reals, Positive and negative real numbers, Closure of set of nonnegative reals, Archimedian property of reals, Density property of reals, Upper and least upper bounds, Existence of least upper bound on a subset of reals, Existence of least upper bound on the subset of reals, Convergence of sequences of real numbers, Rational powers of real numbers, Bounded sequences, Monotone convergence theore, Limit laws, Extended real line, Limits of some general sequences

Unit 3 [12 hours]

Limit points of a sequence, Relation between limits and limit points, Limit superior and limit inferior, Relation between limsup, liminf, sup, and inf, Relation between convergent sequences and their limsup and liminf, Completeness of the set of all real numbers, Squeeze test for convergence of sequences, Subsequences, Relation between (non)convergence of sequences and the existence of convergent subsequences, Relation between limit points and the existence of convergent subsequences, Real powers of real numbers, Finite series, Algebra of finite series, Summation over finite sets (and their unions and intersetctions), Convergence of infinite series as convergence of sequence of partial sums, Zero test. Absolute convergence of infinite series, Absolute convergence implies (conditional) convergence, Alternating series test, Infinite series laws, Comparison test, Convergence criterion for infinite series: Cauchy's condensation criterion, Root test, Ratio test (in terms of limsup and liminf), Convergence of power series (related Riemann zeta function), Raabe's test for absolute convergence of an infinite series, Rearrangement of series

Unit 4 [12 hours]

Open and Closed intervals, Adherernt points, Closure of sets, Relation between of sets, Examples of closure of sets, Limit points, Closure of rationals is reals, Set of all adherent points and closure of a set, Union of a set and its limit points is closure of the set, Limit points as limits of sequences, Bounded sets, Heine-Borel theorem, Convergence of a function at a point, Convergence of a functions as convergence of sequences, Uniqueness of limit of a function at a point, limit laws for functions, Locality of limits, Contnuity of functions, Equivalent definitions of continuity (epsilon-delta, sequential), Arithmatics preserve continuity, left hand and right hand limits, Bounded function, Continuity on closed intervals imply boundedness, Global optima and extreme value theorem, Intermediate value theorem, Uniform continuity, Infinite adherent points, Limits at infinity, Differentiablity at a point, Differentiability implies continuity, Newton's approximation, Differential calculus, Chain rule of differentiation, Local maxima and minima, Staionarity of local maxima/minima, Rolle's theorem, Mean value theorem, L'Hôpital rule