Real Analysis

SYLLABUS- REAL ANALYSIS, Section-A
Dedekind’s definition of real numbers Addition, subtraction and multiplication of
real numbers. Section of sets of real numbers. Lower and upper bounds. Supremum
and infimum of the subsets of R. Completeness of R. (2 questions)
Definition of a sequence. Theorems on limits of sequences. Bounded and monotonic
sequences. Convergence of a sequence. Cauchy’s convergence criteria. Completeness
of R. Bolzano Weierstrass criteria. Limit superior and limit inferior.
Convergence of a series. Series of non-negative terms. The number "e" as an irrational
number. Comparison test. Cauchy’s n th root test. Ratio test. Raabe’s test.
Logarithmic, De Morgan and Bertrand’s tests. Alternating series. Leibnitz test.
Absolute and conditional convergences. (2 questions)
Section-B
Continuous functions and their properties. Classification of discontinuities.
Sequential continuity and limits. Uniform convergence. Differentiability, Chain rule
of differentiability.
Rolle’s theorem. Lagrange’s and Cauchy’s mean value theorems. Darboux theorem.
Indeterminate forms. (2 questions)
Riemann integral. Integrability of continuous and monotonic functions. The
fundamental theorem of Integral Calculus. Mean Value theorems of Integral calculus.
Riemann- Stiltjes integration.
Improper integrals and their convergence Comparison tests. (2 questions)