Real Analysis-II (MAT-202)

Textbooks: 

• Bartle and Sherbert, Introduction to Real Analysis, Fourth Edition, John Wiley & Sons, New York, 2011

• Kumaresan, Topology of Metric Spaces, Narosa Publishing House, New Delhi, 2005

Reference Books: 

• N. L. Carothers, Real Analysis. Cambridge University Press, Cambridge, (South Asia edition) 2009

• W. Rudin, Principles of Mathematical Analysis, third ed., McGraw–Hill, 1976

Topics covered:

Lecture 1:  The algebraic and order properties of the set of real numbers, Absolute value and the real line, its completeness properties

Suggested reading: Bartle & Sherbert, Sections 2.1, 2.2, 2.3

Lecture 2: Basic notions of sequence and its convergence and boundedness

Suggested reading: Bartle & Sherbert, Section 3.1.

Lecture 3: Algebra of limits of a sequence and using it to check the convergence of a sequence

Suggested reading: Bartle & Sherbert, Section 3.2

Lecture 4:  Monotone sequences, monotone convergence theorem and its applications

Suggested reading: Bartle & Sherbert, Section 3.3

Lecture 5: Subsequences, the Bolzano-Weierstrass Theorem

Suggested reading: Bartle & Sherbert, Section 3.4

Homework 1 (due Aug 21): Bartle & Sherbert, Section 3.3: 1, 2; Section 3.4: 8, 10, 11, 12, 13.

Lecture 6: Cauchy criteria for convergence

Suggested reading: Bartle & Sherbert, Section 3.5

Lecture 7: Basic notions of series of a sequence and Cauchy criteria for convergence of series

Suggested reading: Bartle & Sherbert, Section 3.7

Homework 2 (due Aug 28): Bartle & Sherbert, Section 3.7:  2, 3, 6, 9, 14, 15

Lecture 8: Basic notion of continuity of functions, algebra of continuous functions

Suggested reading: Bartle & Sherbert, Sections 5.1, 5.2

Lecture 9: Sequential criteria of continuity of functions, maximum-minimum theorem

Suggested reading: Bartle & Sherbert, Sections 5.1, 5.2

Homework 3 (due Aug 31): Bartle & Sherbert, Section 5.1: 10, 12; Section 5.2: 7, 10, 12, 15.