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.
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.