# Real Analysis

By Prof. Surajit Borkotokey   |   Dibrugarh University
Learners enrolled: 536
This course on Real Analysis is basically prepared for the Core course C 3.1 of the UGC prescribed syllabus on Mathematics non-Honours, however, it can be taken by learners with Mathematics Honours complementing to the Core course C 2.1 or even as a GE course by the learners with Honours in Physics, Economics, Statistics etc.
This course is designed to provide the learners an understanding of the analytical aspects of various mathematical objects pertaining to the set of real numbers. It covers the basic characterizing properties of real numbers, the sequence and series of real numbers and their convergence and finally, the sequence of real valued functions and their convergence. For pursuing any course on Pure and applied Mathematics or from other disciplines that involve analytical aspects of the Mathematical objects, this course is a must.

Summary
 Course Status : Completed Course Type : Core Duration : 12 weeks Category : Mathematics Credit Points : 6 Level : Undergraduate Start Date : 23 Jan 2023 End Date : 16 Apr 2023 Enrollment Ends : 15 Mar 2023 Exam Date :

### Course layout

Unit 1:
Finite and infinite sets, examples of countable and uncountable sets. Real line, field properties of R, bounded sets, suprema and infima, completeness property of R, Archimedean property of R, intervals. Concept of cluster points and statement of Bolzano-Weierstrass theorem.
Unit 2:
Real Sequence, Bounded sequence, Cauchy convergence criterion for sequences. Cauchy’s
theorem on limits, order preservation and squeeze theorem, monotone sequences and their convergence, monotone convergence.
Unit 3:
Infinite series. Cauchy convergence criterion for series, positive term series, geometric series,
comparison test, convergence of p-series, Root test, Ratio test, alternating series, Leibnitz’s test. Definition and examples of absolute and conditional convergence.
Unit 4:
Sequences and series of functions, Pointwise and uniform convergence. Mn-test, M-test,
Statements of the results about uniform convergence and integrability and differentiability of functions, Power series and radius of convergence.

### Books and references

1. T. M. Apostol, Calculus (Vol. I), John Wiley and Sons (Asia) P. Ltd., 2002.
2. R.G. Bartle and D. R Sherbert, Introduction to Real Analysis, John Wiley and Sons (Asia) P.Ltd., 2000.
3. E. Fischer, Intermediate Real Analysis, Springer Verlag, 1983.
4. K.A. Ross, Elementary Analysis- The Theory of Calculus Series- Undergraduate Texts in Mathematics, Springer Verlag, 2003.
5.     A. Kumar and S. Kumaresan, A Basic Course in Real Analysis, CRC Press. 2020.

5. http://www.math.louisville.edu/~lee/RealAnalysis/

6. https://artofproblemsolving.com/community/c7t430f7_real_analysis_theorems

7. http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF

### Prof. Surajit Borkotokey

Dibrugarh University
Prof. Surajit Borkotokey has 23 years of experience in PG teaching. Currently he teaches Real Analysis and Measure Theory in MA/M.Sc. Program in his University.
He has been associated with teaching of Mathematics at under-graduate level at different institutions. His area of research is cooperative game theory and Aggregation Operators. An Indo-US fellow, Prof Borkotokey has been a visiting professor at Louisiana State University, USA; Beijing Institute of technology, China and Slovak University of Technology, Slovakia.

### Course certificate

1. Internal Assessment : 30%
2. External Assessment: 70%

Note 1. Internal Assessment includes Assignments to be submitted online by the learners, Interactions with the learners and time to time online MCQs.
Note 2.  External Assessment includes end-term examinations.

Minimum 40% in each would be required to pass the course and get completion certificate.