Description

This is the first course in analysis. It develops the fundamental ideas of analysis and is aimed at developing the student’s ability in reading and writing mathematical proofs of theorems that meet rigorous standards based on content, organization and coherence, argument and support, and style and mechanics. Another objective is to provide sound understanding of the axiomatic foundations of the real number system, in particular the notions of completeness and compactness.

LEARNING OUTCOMES

1. Describe the real line as a complete, ordered field,
2. Determine the basic topological properties of subsets of the real numbers,
3. Use the definitions of convergence as they apply to sequences, series, and functions,
4. Determine the continuity and differentiability of functions defined on subsets of the real line,
5. Apply the Mean Value Theorem to problems in the context of real analysis, and
6. Produce rigorous proofs of results that arise in the context of real analysis.

COURSE CONTENTS

1. Number Systems: Ordered fields,
2. rational, real and complex numbers,
3.  Archimedean property,
4. supremum, infimum and completeness.
5. Topology of real numbers:
6.  Convergence, completeness, completion of real numbers,
7. open sets, closed sets, compact sets,
8.  Heine Borel theorem, connected sets.
9. Sequences and Series of Real Numbers:
10. Limits of sequences, algebra of limits.
11.  Bolzano Weierstrass theorem, Cauchy sequences, liminf, limsup,
12. Limits of series, convergences tests, absolute and conditional convergence, power series.
13. Continuity: Functions, continuity and compactness, existence of minimizers and maximizers,
14. uniform continuity, continuity and connectedness, intermediate mean value theorem,
15. Monotone functions and discontinuities.
16. Differentiation: Mean value theorem, L’Hopital’s Rule, Taylor’s theorem.          

Recommened Books

1. Bartle R. G.  and Sherbert D. R. Introduction to Real Analysis. John Wiley & Sons, Inc 2011.
2. Lang S., Analysis I, Addison-Wesley Publ. Co., Reading, Massachusetts, 1968.
3. Rudin W., Principles of Mathematical Analysis, 3rd Ed., McGraw-Hill, 1976.
4. Habibullah G. M., Real Analysis, Ilmi Kitab Khana, Lahore, Pakistan, 2002.
5. Royden H.L, FitzPatrick P.M. Real Analysis, 4th ed, 2009

Assessment Criteria

       Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

       Mid-Term Exam:   30

       Final-Term Exam: 50

Key Dates and Time of Class Meeting

Wednesday                                             2:00 AM-3:30 PM (SS)

Thursday                                                 2:00 AM-3:30 PM (SS)

Thursday                                                 9:30 AM-11:00 PM (Reg)

Friday                                                       2:00 AM-3:30 PM (Reg)

Commencement of Classes                                                   October 12, 2020

Mid Term Examination                                                            December 14-18, 2020

Final Term Examination                                                          February 08-12, 2021

Declaration of Result                                                              Februray 19, 2021