THIRD YEAR

1.3: Finite and Infinite Sets- definition, countable sets, denumerability of Q, union of countable sets, Cantor’s theorem.
2.1: The Algebraic and Order Properties of R- algebraic properties, basic results , rational and irrational numbers, irrationality of √2, Order properties , arithmetic-geometric inequality, Bernoulli’s Inequality.
2.2: Absolute Value and the Real Line- definition, basic results, Triangle Inequality, The real line, ε-neighborhood.
2.3: The Completeness Property of R- Suprema and Infima, alternate formulations for the supremum , The Completeness Property.
Module-II
2.4: Applications of the Supremum Property- The Archimedean Property, various consequences , Existence of √2, Density of Rational Numbers in R, The Density Theorem, density of irrationals.
2.5: Intervals-definition, Characterization of Intervals, Nested Intervals, Nested Intervals Property, The Uncountability of R, [binary, decimal and periodic representations omitted] Cantor’s Second Proof.
3.1: Sequences and Their Limits- definitions, convergent and divergent sequences, Tails of Sequences, Examples.
3.2: Limit Theorems- sum, difference, product and quotients of sequences , Squeeze Theorem, ratio test for convergence.
3.3: Monotone Sequences- definition, monotone convergence theorem , divergence of harmonic series , calculation of square root, Euler’s number.
Module-III
3.4: Subsequences and the Bolzano-Weierstrass Theorem - definition, limit of subsequences, divergence criteria using subsequence , The Existence of Monotone Subsequences, monotone subsequence theorem, The Bolzano-Weierstrass
Theorem, Limit Superior and Limit Inferior.
3.5: The Cauchy Criterion - Cauchy sequence, Cauchy Convergence Criterion, applications, contractive sequence.
3.6: Properly divergent sequences- definition, examples, properly divergent monotone sequences, ‘‘comparison theorem’’ , ‘‘limit comparison theorem’’.
11.1: Open and Closed sets in R - neighborhood, open sets, closed sets, open set properties, closed set properties , Characterization of Closed Sets, cluster point , Characterization of Open Sets, The Cantor Set, properties.
Module-IV
1.1: Complex numbers and their properties- definition , arithmetic operations, conjugate, inverses, reciprocal.
1.2: Complex Plane- vector representation , modulus, properties , triangle inequality.
1.3: Polar form of complex numbers- polar representation, principal argument, multiplication and division, argument of product and quotient , integer powers, de Moivre’s formula.
1.4: Powers and roots- roots, principal n th root.
1.5: Sets of points in the complex plane- circles, disks and neighbourhoods, open sets, annulus, domains, regions, bounded sets.
2.1: Complex Functions- definition, real and imaginary parts of complex function, complex exponential function, exponential form of a complex number, Polar Coordinates.
2.2: Complex Functions as mappings- complex mapping , illustrations , Parametric curves in complex planes, common parametric curves, image of parametric curves under complex mapping .