Real Analysis

Unit- 1

Review of Algebraic and Order Properties of R, neighborhood of a point in R. Bounded above sets, Bounded below sets, Bounded Sets, Unbounded sets, Suprema and Infima, The Completeness Property of R, The Archimedean Property, Existence of greatest integer function, Density of Rational (and Irrational) numbers in R, Intervals, Nested interval theorem, Limit points of a set, Isolated points, Illustrations of Bolzano- Weierstrass theorem for sets. Idea of finite sets, countable sets,

uncountable sets and uncountability of R.

Unit- II

Sequences, Bounded sequence, Convergent sequence, Limit of a sequence. Limit Theorems, Monotone Sequences, Monotone Convergence Theorem. Subsequences, Divergence Criteria, Monotone Subsequence Theorem (statement only), Bolzano Weierstrass Theorem for Sequences. Cauchy sequence, Cauchy’s Convergence Criterion. Infinite series and its convergence, Cauchy Criterion.

 (1) End-Term Examination:

o    Weightage: 70% of the final result

o    Minimum Passing Criteria: 40% 

 (2) Internal Assessment:

o    Weightage: 30% of the final result

o    Minimum Passing Criteria: 40% 

ü  Calculation of IA Marks:

o    Out of all graded weekly assessments/assignments, the top 50% (equal-weighted) shall be considered for calculation of the final Internal Assessment marks.

ü  Weekly Assignments- Each week of the course shall include one MCQ-based assessment. The availability of the weekly assignments will be posted along with the assignments themselves.