This course in real analysis aims to teach students the concepts of limit theorems, continuity of functions, sets of real numbers, metric spaces, compact sets, sequences, series, and theorems related to convergence and divergence. Students will learn about open sets, closure of sets, ordered sets, least upper bound, greatest lower bound, Cauchy sequences, infinite series, comparison tests for series, one-sided limits, continuity properties, boundedness theorem, and types of discontinuities. The course employs a tutorial format and covers topics such as Weierstrass Theorem, Heine Borel Theorem, connected sets, and various tests for convergence. The intended audience for this course includes individuals interested in gaining a foundational understanding of real analysis concepts and principles.
Overview
Syllabus
Limit Theorems for functions (CH_30).
Continuity of Functions (Ch-30).
Finite, Infinite , Countable and Uncountable Sets of Real Numbers.
Types of Sets with Examples,Metric Space.
Various properties of open set, closure of a set.
Ordered set, Least upper bound, greatest lower bound of a set.
Compact Sets and its properties.
Weiersstrass Theorem, Heine Borel Theorem,Connected set.
Tutorial II.
Concept of limit of a sequence.
Some Important limits, Ratio tests for sequences of Real Numbers.
Cauchy theorems on limit of sequences with examples.
Theorems on Convergent and Divergent sequences.
Cauchy sequence and its properties.
Infinite series of real numbers.
Comparision tests for series, Absolutely convergent and Conditional Convergent series.
Tests for absolutely convergent series.
Raabe's test, limit of functions, Cluster point.
Some results on limit of functions.
Limit Theorems for Functions.
Extension of limit concept (One sided limits).
Continuity of Functions.
Properties of Continuous functions.
Boundedness theorem, Max-Min Theorem and Bolzano's theorem.
Uniform continuity and Absolute continuity.
Types of Discontinuities, Continuity and Compactness.
Continuity and Compactness (Contd.) Connectedness.
Continuum and Exercises.
Equivalence of Dedekind and Cantor's Theory.
Irrational numbers, Dedekind's Theorem.
Rational Numbers and Rational Cuts.
Cantor's Theory of Irrational Numbers (Contd.).
Cantor's Theory of Irrational Numbers.
Continuum and Exercises (Contind..).
Taught by
Ch 30 NIOS: Gyanamrit
