Explore a comprehensive course on real analysis covering fundamental concepts and theorems. Delve into limit theorems, continuity of functions, and set theory, including finite, infinite, countable, and uncountable sets of real numbers. Examine metric spaces, open sets, and their properties. Study ordered sets, least upper bounds, and greatest lower bounds. Investigate compact sets, the Weierstrass Theorem, and the Heine-Borel Theorem. Learn about connected sets and explore sequences, including convergence, divergence, and Cauchy sequences. Analyze infinite series, convergence tests, and absolute convergence. Discover limits of functions, cluster points, and continuity concepts. Examine properties of continuous functions, including boundedness and max-min theorems. Investigate uniform and absolute continuity, types of discontinuities, and the relationship between continuity, compactness, and connectedness. Explore the equivalence of Dedekind and Cantor's theories, irrational numbers, and rational cuts. Gain a solid foundation in real analysis through lectures, tutorials, and exercises over the course of 1 day and 7 hours.
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
