MTH621 Official Handouts

About the instructor

Dr. Malik did his MS and PhD (Mathematics) from University of La Rochelle, La Rochelle, France in 2009 and 2012, respectively. Prior to MS and PhD, Dr. Malik completed his MPhil and MSc (Mathematics) from Department of Mathematics, University of the Punjab, Lahore, Pakistan. He has been a liated with several universities in Pakistan and abroad. He has the experience of teaching a wide range of mathematics courses at undergraduate and graduate level.

Dr. Malik has published several research articles in international journals and conferences. His area of research includes the study of dierential equations with nonlocal operators and their applications to image processing. He is also interested in inverse problems related to reaction-diusion equations with nonlocal integrodierential operators and boundary conditions. These models have numerous applications in anomalous diusion/transport, biomedical imaging and non-destructive testing.

About the handouts

The books followed during this course are: W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, 1976. ISBN: 9780070542358. and Consequently, the most of the examples considered in these notes are from the above mentioned books and their exercises, but not restricted to those books only. If you nd any typing error in the text kindly report to me by writing an email to

Course Information

Title and Course Code:

Number of Credit Hours: 3 credits

Course Objective: The Real Analysis I is the rst course towards the rigorous (formal) treatment of the fundamental concepts of mathematical analysis. This course could be considered as the fundamental course in pursue of mathematical study at undergraduate or master level. Although, the topics of the Real Analysis I are self contained but someone having knowledge of Calculus (single and multivariable) and Dierential Equations will be comfortable with the contents of the course. These subjects could be considered as prerequisite of the Real Analysis I. Learning Outcomes (for the whole course) Upon completion of this course students will be able to

• Understand the set theoretic statements, the real and complex number sys- tems.
• Apply principle of mathematical induction, ordered sets.
• Decide about the convergent or divergent sequences and series.
• De ne the limit of a function, prove various theorems about limits, sequences and functions.
• Check the continuity of real valued functions, prove various theorems about continuous functions with emphasize on the proofs.
• Understand the derivative of a function, proof of various theorems about dif- ferentiability of the function (LO6).
• Prove and apply Bolzano-Weierstrass theorem, Mean value theorem.
• De ne Riemann integral, Riemann sums, proof of various results about the Riemann integrals.

Prerequisites: Calculus with Analytical Geometry

The textbooks for this course:

[1] W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, 1976. ISBN: 9780070542358.

Reference books:

Contents

1. 1 Basic Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Universal Set . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Set Builder Notion . . . . . . . . . . . . . . . . . . . . . . . .
2. 2 Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. 3 Principle of Mathematical Induction 1.3.1 An initial segment
4. 4 Finite and In nite Set . . . . . . . . . . . . . . . . . . . . . . . . . .
5. 5 Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6. 6 Dedekind In nite Set . . . . . . . . . . . . . . . . . . . . . . . . . . .
7. 7 The Set of Rational Numbers . . . . . . . . . . . . . . . . . . . . . .
8. 8 Ordered Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Partial Order . . . . . . . . . . . . . . . . . . . . . . . . . . .
9. 9 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 In mum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10. 10 Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.1 Supremum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11. 11 Least Upper Bound Property
12. 12 The Completeness Axiom
13. 13 The Archimedean Property of R
14. 14 Dense Set in R
15. 15 The Set of Rational Numbers is not Complete . . . . . . . . . . . . .
16. 16 Applications of Properties of Field
17. 17 The Extended Real Number System
18. 18 Principle of Mathematical Induction 1.18.1 Principle of Mathematical Induction For Z . . . . . . . . . . .
19. 19 Generalization of Union and Intersection . . . . . . . . . . . . . . . .
20. 20 Open Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21. 1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Convergent Sequence . . . . . . . . . . . . . . . . . . . . . . .
22. 2 Bounded Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23. 3 Monotonic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . .
24. 4 Some Special Sequences
25. 5 Subsequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26. 6 Limit Superior and Limit Inferior . . . . . . . . . . . . . . . . . . . .
27. 7 Cauchy Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28. 8 Series