Course Tittle: REAL ANALYSIS

Course Code: MATH-307

Credit Hours: 03

Instructor: WAJIHA PERVAIZ

Email: [email protected]

                                                                             DESCRIPTION & OBJECTIVES:-

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

   Recommended Books:

       1. Lang S., Analysis I, Addison-Wesley Publ. Co., Reading, Massachusetts, 1968.

        2. Rudin W., Principles of Mathematical Analysis, 3rd Ed., McGraw-Hill, 1976.

       3. Habibullah G. M., Real Analysis, Ilmi Kitab Khana, Lahore, Pakistan, 2002.

        4. Royden H.L, FitzPatrick P.M. Real Analysis, 4th ed, 2009

                                                                                                                    CONTENTS:-

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, lim inf, lim sup, limits of series, convergences tests, absolute and

conditional convergence, power series. Continuity: Functions, continuity and compactness, existence of minimizers and maximizers, uniform continuity, continuity

and connectedness, intermediate mean value theorem, monotone functions and discontinuities. Differentiation: Mean value theorem, L’Hopital’s Rule, Taylor’s

theorem.

RESEARCH PROJECT:-

N/A

ASSESSMENT CRITERIA:-

Mid exam: 30

Sessional: 20

Project: --

Assignments: 10

Presentation: 10

Final exam: 50

Total: 100

RULES AND REGULATIONS:-

75% attendance is compulsory to appear in Final Term exam.