21MAT307 REAL ANALYSIS MATHEMATICS PAPER – V-

Unit I-

Review: Sets and Functions – Mathematical Induction – Finite and Infinite Sets.
The Real Numbers: The Algebraic and Order Properties of R – Absolute Value and the Real Line – The Completeness Property of R – Applications of the Supremum Property – Intervals.

Unit II-

Sequences and Series: Sequences and Their Limits – Limit Theorems – Monotone Sequences – Subsequences and the Bolzano-Weierstrass Theorem – The Cauchy Criterion – Properly Divergent Sequences – Introduction to Infinite Series – Absolute Convergence of Infinite series – Tests for Absolute convergence – Tests for Non-absolute convergence.

Unit III-

Limits and Continuous Functions: Limits of Functions – Limit Theorems – Some Extensions of the limit concept – Continuous Functions – Combinations of Continuous Functions – Continuous Functions on Intervals – Uniform Continuity.

Unit IV-

Differentiation: The Derivative – The Mean Value Theorem – L’Hospital’s Rules – Taylor’s Theorem.

Unit V-

The Riemann Integral: Riemann Integral – Riemann Integrable Functions – The Fundamental Theorem – Approximate Integration.

TEXTBOOK:

1. Robert Gardner Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th Edition, John Wiley & Sons, 2011.

REFERENCES:

1. Tom M. Apostol, Mathematical Analysis, 2nd Edition, Narosa publishing house, New Delhi,1989.
2. Rudin. W, Principles of Mathematical Analysis, 3rd Edition, McGraw-Hill International Editions, 1976. H.L. Royden and P.M. Fitzpatrick, Real Analysis, 4th Edition. Pearson Education Asia Limited, 2010.