Course: MAT 376 3.0 Real Analysis II (Compulsory)
Course Content: Properties of the real numbers:
Set theoretic preliminaries, Real number system as a complete ordered field, Sequences of real numbers, Subsequences, Bounded and monotone sequences, Convergence and the Cauchy criterion, Limit superior and limit inferior for sequences, Some special sequences, Series of real numbers, Serries convergence, Absolute and conditional convergence, tests for convergence, Rearrangements of series, Real functions, Boundedness and monotonicity, Limits and continuity at a point, Continuity on an interval, Intermediate value theorem andextreme value theorem, Uniform continuity, Limits at infinity and infinite limits, Differentiability and the derivative of a real function, Rolle’s and mean value theorems, Higher order derivatives, Sequences and series of functions, Uniform and point-wise convergence, Weierstrass M-test, Uniform convergence and continuity, Uniform convergence and differentiability, Power series and radius of convergence, Exponential and logarithmic functions, Trigonometric functions.
Recommend Readings:
- Rudin, W., The Principles of Mathematical Analysis, 3rd International Series in Pure & Applied Mathematics, 2006
- Bartle, R. G., & Sherbert, D. R., Introduction to Real Analysis, 4th Wiley 2011
- Gooldberg, R. T., Methods of Real Analysis, 2nd Wiley, 1976
- Pugh, C. C., Real Mathematical Analysis, 2nd Undergraduate Texts in Mathematics, 2015
- Mattuck, A., Introduction to Analysis, Illustrated ed. Prentice Hall, 1999
- Hammack, R., BOOK OF PROOF, 3rd Richard Hammack, 2018 (https://www.people.vcu.edu/~rhammack/BookOfProof/)