This lecture series provides a comprehensive introduction to real analysis, drawing on Terence Tao's Analysis I. Structured to build from foundational concepts to advanced theorems, it begins with the Axioms of Completeness and Least Upper Bound Property, exploring real number structure, sequences (convergence, limits, Cauchy sequences), and key results like Bolzano-Weierstrass. The series transitions to continuity (epsilon-delta definitions, Intermediate/Extreme Value Theorems), uniform continuity, and algebraic properties of continuous functions. Later lectures focus on differentiation (derivatives, Mean Value Theorem) and applications including Inverse Function Theorem and L’Hôpital’s Rule. Emphasizing rigorous proofs and central theorems, the series connects theoretical foundations with analytical tools, offering systematic progression from real numbers to function behavior while highlighting Tao's pedagogical approach.
| Lecture | Topics Covered | Speaker | Resources (Notes/Problems) |
|---|---|---|---|
| Lecture 1 May 7 (Wed) |
Introduction to Sequences: Definition of a sequence, convergence, limits, uniqueness of limits, bounded sequences. | E.K. Narayanan TA: Renjith T. |
Problem Sheet |
| Lecture 2 May 8 (Thu) |
Limit Laws, Monotonicity, and Cauchy Sequences: Limit laws, monotonic sequences (definition, examples), Cauchy sequences, completeness via Cauchy. | E.K. Narayanan TA: Renjith T. |
Problem Sheet |
| Lecture 3 May 12 (Mon) |
Subsequences and Bolzano-Weierstrass: Subsequences, convergence of subsequences, Bolzano-Weierstrass Theorem. | Jayanthan A.J. TA: Renjith T. |
Problem Sheet |
| Lecture 4 May 13 (Tue) |
Defining Continuity: The epsilon-delta definition of continuity, the sequential criterion for continuity. | Jayanthan A.J. TA: Renjith T. |
Problem Sheet |
| Lecture 5 May 14 (Wed) |
Examples and Counterexamples of Continuity: Detailed analysis of continuity in functions (e.g., polynomials, piecewise functions), common discontinuities. | Jayanthan A.J. TA: Renjith T. |
Problem Sheet |
| Lecture 6 May 19 (Mon) |
Intermediate and Extreme Value Theorems: The Intermediate Value Theorem for continuous functions, the Extreme Value Theorem on closed intervals. | Manjunath Krishnapur TA: Renjith T. |
Coming Soon |
| Lecture 7 May 22 (Thu) |
Algebra of Continuous Functions: Continuity of sums, differences, products, quotients, and compositions of continuous functions. | Manjunath Krishnapur TA: Renjith T. |
Problem Sheet |
| Lecture 8 May 23 (Fri) |
Uniform Continuity and Compactness: Definition of uniform continuity, contrast with pointwise continuity, uniform continuity on compact sets. | Manjunath Krishnapur TA: Renjith T. |
Coming Soon |
| Lecture 9 May 27 (Tue) |
Introduction to Differentiation: Definition of the derivative, differentiability implies continuity, geometric interpretation of the derivative. | Manjunath Krishnapur TA: Renjith T. |
Coming Soon |
| Lecture 10 May 28 (Wed) |
Differentiation Rules and Local Extrema: Basic differentiation rules (sum, product, quotient, chain rule), definition of local maxima and minima, Fermat's Theorem. | Sanjay P. K. TA: Renjith T. |
Coming Soon |
| Lecture 11 May 29 (Thu) |
Mean Value Theorem and Applications: Rolle's Theorem, the Mean Value Theorem, and applications including using the derivative to determine monotonicity. | Sanjay P. K. TA: Renjith T. |
Problem Sheet |
| Lecture 12 May 30 (Fri) |
Inverse Function Theorem and L'Hôpital's Rule: Differentiability of inverse functions and the Inverse Function Theorem, L'Hôpital's Rule for evaluating indeterminate forms. | Sanjay P. K. TA: Renjith T. |
Coming Soon |