====== MATH-305: Real Analysis-I ======

=====Objectives of the course:=====
This is the first rigorous course in analysis and has a theoretical emphasis. It tegorously develops the fundamental ideas of calculus and is aimed to develop the students’ ability to deal with abstract mathematics and mathematical proofs.

=====Course Contents:=====
Real Number System: Ordered fields, The field of Reals, The extended real number system, Euclidean space.
Numerical Sequences and Series: Limit of a sequence, Bounded sequence, Monotone sequences, Limit superior and inferior, Subsequences, Infinite series of constants, Test for convergence of series, Absolute and conditional convergence.
Continuity: Limit of a function and continuous function, Continuity and compactness, Continuity and connectedness, Uniform continuity, Kind of discontinuities.
Differentiation: The derivative of a real function, Mean value theorems, The continuity of   derivatives, Taylor’s theorem.     
Riemann Stieltjes Integral: Definition of Riemann Integral, Upper and lower sums, Integrability criterion, Classes of integrable functions, Properties of the Riemann Integral.
=====Books Recommended:===== 
  - Rudin W. 1976. Principles of Mathematical Analysis. 3rd  ed. MCGraw Hill.
  - Apostal T.M., 1974. Mathematical Analysis. 2nd ed. Addison Wesley.
  - Kaplan W., 1973. Advanced calculus. 2nd ed.  Addison Wesley.
  - Rabenstein R.L.,1984. Elements of Ordinary differential equations. 1st ed. Academic Press.
  - Bartle R,G, Donald R.S., 1999. Introduction to Real Analysis, 3rd ed. Wiley.
  - Royden H. 1988. Real Analysis. 3rd ed. Prentice Hall/ Pearson Edition.