Unit 1

Sets and Functions, surjective and injective functions, inverse functions. Countable and Uncountable sets, Countability of Q, Absolute value and the Real line, Completeness property of R, Least upper bound property and its applications, Nested intervals, Cantor’s proof of uncountability of R. (Sections 1.1, 1.3, 2.2, 2.3, 2.4, 2.5)

Unit 2

Sequences and Their Convergence, Monotone Sequences, Monotone Convergence Theorem, Subsequences and Bolzano-Weierstrass Theorem, Cauchy Sequence and Cauchy Convergence criterion. (Sections 3.1 to 3.5)

Unit 3

Continuity, Uniform Continuity. Derivative of functions, Mean Value Theorem, L’Hospital Rule, Taylor’s Theorem. Infinite Series: Conditional and Absolute Convergence, Tests for Absolute Convergence of Infinite Series, Alternating Series Test, Rearrangement of terms in an infinite Series.

Unit 4

Riemann Integration: Integral and its properties, Fundamental theorems of Calculus, Sum of an infinite series as an integral, Improper Riemann integrals. (Sections 7.1 to 7.3)

Unit 5

Open and Closed Sets in R, Characterization of Open and Closed Sets, Compact Sets, Heine-Borel Theorem. (Sections 11.1, 11.2)

Course Outcome:
CO1: Understanding the set theoretic statements and the completeness property of R.
CO2: Understanding the concepts of sequences, series and Limits. Apply the tests for convergence, absolute convergence and analyzing the convergence criteria.
CO3: Defining Limits, continuity and monotonicity of a function and understanding the theorems related to them.
CO4: Understanding the concepts of extreme values, Mean value theorem and applying Taylor’s theorem for approximating functions.
CO5: Understanding Riemann Sum and apply it to approximate integrations

Textbook:
Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis, 3rd edition, Wiley.

References:

1. S. Kumaresan and Ajit Kumar, A Basic Course in Real Analysis, CRC Press.
2. V.K. Krishnan, Fundamentals of Real Analysis, Pearson. Terence Tao, Analysis I, Hindustan Book Agency. Terence Tao, Analysis II, Hindustan Book Agency.