Specific Objectives of course:

This is the first course in analysis. It develops the fundamental ideas of analysis and is aimed at developing the  students’  ability  in  reading  and  writing  mathematical  proofs. Another objective is to provide sound understanding of the axiomatic foundations of the real number system, in particular the notions of completeness and compactness.

Course Outline:


Number Systems:

Ordered fields. Rational, real and complex numbers. Archimedean property, supremum, infimum and completeness.

Topology of real numbers:

Convergence, completeness, completion of real numbers. Open sets, closed sets, compact sets. Heine Borel Theorem. Connected sets.

Sequences  and  Series  of  Real  Numbers:

Limits  of  sequences, algebra of limits. Bolzano Weierstrass Theorem. Cauchy sequences, liminf, limsup. Limits of series, convergences tests, absolute and conditional convergence. Power series.


Functions, continuity and compactness, existence of minimizers and maximizers, uniform continuity. Continuity and connectedness, Intermediate mean Value Theorem. Monotone functions and discontinuities.


Mean Value Theorem, L’Hopital’s Rule, Taylor’s Theorem.