# Real Analysis (RA-101A)

Most content of the course is based on the Terence Tao lectures on real analysis and his books.

Contents:

CH-1-Introduction

CH-2-Natural Numbers

$\bullet \text{ Peano's Axiom} \; \bullet \textbf {Euclid Algorithm}$

CH-3- Set Theory

$latex \bullet \text{ Axiom of specification} \; \bullet \textbf { Russell’s Paradox}$

CH-4- Rational Number and Integers

$\bullet \text{ Integer as Commutative Ring and Ring} \; \bullet \text { Rational as Field and Ordered Field}$

CH-5- Real Numbers

$\bullet \textbf{ Cauchy Sequence} \; \bullet \text { Bounded Sequence} \; \bullet \text{ Equivalent Cauchy Sequences} \; \bullet \text { Construction of Real Numbers} \; \bullet \text { Ordering of Reals}$

CH-6- Limits of Sequences

$\bullet \text{ Cauchy Sequences of Reals} \; \bullet \text { Convergence of Real seq} \; \bullet \text{ Uniqueness of limits} \; \bullet \text { limits of seq} \; \bullet \text { Laws of Limits} \; \bullet \text{ Supremum and Infimum } \; \bullet \text {Monotone Bounded seq converge } \; \bullet \text{Limit Points } \; \bullet \text {Limit Superior and Limit Inferior } \; \bullet \text {Comparison Principle} \; \bullet \text{ Squeeze test} \; \bullet \text { Zero test for seq} \; \bullet \text{ Completeness of reals} \; \bullet \text{ Subsequences and Its theorems} \; \bullet \textbf {Bolzano-Weistrass theorem } \; \bullet \text{Compactness on real lines }$

CH-7- Series

$\bullet \text { Finite series (triangle inequality, comparison test, summation, linearity, monotonicity) } \; \bullet \textbf { Fubini's Theorem for finite series } \; \bullet \text{Infinite Series (Convergence, Absolute and Conditional Convergence) } \; \bullet \text {Absolute Convergence test } \; \bullet \text { Alternating series test } \; \bullet \text { Infinite series law } \; \bullet \text{ telescoping series } \; \bullet \text { Sums of Non-Negative numbers (Comparison test, contrapositive) } \; \bullet \text { Geometric Series} \; \bullet \text {Cauchy criteria for a decreasing sequence to converge} \; \bullet \textbf {Harmonic series and Riemann-zeta function } \; \bullet \text{Rearrangement of Infinite series } \; \bullet \text {Root and Ratio tests }$

format:

$\; \bullet \text { } \; \bullet \text { } \; \bullet \text{ } \; \bullet \text { }$

Materials:

Books: Analysis I and Analysis II.