REAL ANALYSIS

1. Axiomatic Study of Real Numbers

(Completeness Property in R, Archimedian Property, 

Countable and Uncountable Sets)

2. Neighborhood, Interior Points, Limit Points, Open and Closed

Sets, Derived Sets, Dense Sets, Perfect Sets, Bolzano-Weierstrass

Theorem  

3. Sequences of Real Numbers, Sub-Sequences, Bounded and 

Monotonic Sequences, Convergent Sequences, Cauchy's Theorems 

on Limits, Cauchy's General Principle of Convergence

4. Uniform Convergence of Sequences and Series of Functions,

Weierstrass M-Test, Abel's and Dirichlet's Tests

5. Sequential Continuity, Boundedness and Intermediate Value

Properties of Continuous Functions, Uniform Continuity, 

Meaning of Sign of Derivative, Darboux Theorem 

6. Limit and Continuity of Functions of Two Variables

7. Taylor's Theorem for Functions of Two Variables

8. Maxima and Minima of Functions of Three Variables, Lagrange's 

Method of Undetermined Multipliers

9. Riemann Integral, Integrability of Continuous and Monotonic 

Function, the Fundamental Theorem of Integral Calculus, Mean

Value Theorems of Integral Calculus

10.The Riemann-Stieltjes Integral 

11. Improper Integrals and their Convergence, Comparison Test,

µ-test, Abel's Test, Dirichlet's Test

12. Metric Spaces, Neighborhoods, Interior Points, Limit Points, 

Open and Closed Sets, Subspaces

13. Convergent and Cauchy Sequences, Completeness, Cantor's 

Intersection Theorem