DESCRIPTION 

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.

LEARNING OUTCOMES

After the completion of the course, students will be able to

1. Describe fundamental properties of the real numbers that lead to the formal development of real analysis.
2. Comprehend rigorous arguments developing the theory underpinning real analysis.
3. Demonstrate an understanding of limits and how they are used in sequences, series, differentiation and integration.
4. Construct rigorous mathematical proofs of basic results in real analysis.
5. Appreciate how abstract ideas and rigorous methods in mathematical analysis can be applied to important practical problems.

Contents

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

Recommended Books

1. Walter Rudin, Principles of Mathematical Analysis, (3rd Ed, 1976)
2. T.M. Apostal, Mathematical Analysis, (Addison Wesley, 1957)

Suggested Books

1. Halsey Royden, Real Analysis, (3rd Ed. Prentice Hall, 1988)
2. H.L. Royden,Real Analysis,(3rd Ed, 1989)
3. S. Lang, Analysis I, (Addison-Wesley Publ. Co., Reading, Massachusetts, 1968.)

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

Monday                                          9:30 AM-11:00 AM (Reg)

Friday                                              11:00 AM-12:30 PM (Reg)

Commencement of Classes                                                   October 26, 2020

Mid Term Examination                                                            December 28 to January 1, 2020

Final Term Examination                                                          March 01-05, 2021

Declaration of Result                                                              March 12, 2021