InfoCoBuild

Introductory Course in Real Analysis

Introductory Course in Real Analysis. Instructor: Prof. P.D. Srivastava, Department of Mathematics, IIT Kharagpur. This is a basic course in Real Analysis which is a back bone of any course on pure and applied Mathematics and Statistics. This is a very useful course for any branch of science and engineering. The present course has been designed to introduce the subject to undergraduate/postgraduate students in science and engineering. The course contains a good introduction to each topic and an advance treatment of theory at a fairly understandable level to the students at this stage. Each concept has been explained through examples and application oriented problems. (from nptel.ac.in)

Lecture 13 - Heine-Borel Theorem

Go to the Course Home or watch other lectures:

Lecture 01 - Countable and Uncountable Sets

Lecture 02 - Properties of Countable and Uncountable Sets

Lecture 03 - Examples of Countable and Uncountable Sets

Lecture 04 - Concepts of Metric Space

Lecture 05 - Open Ball, Closed Ball, Limit Point of a Set

Lecture 06 - Tutorial I

Lecture 07 - Some Theorems on Open and Closed Sets

Lecture 08 - Ordered Set, Least Upper Bound, Greatest Lower Bound of a Set

Lecture 09 - Ordered Set, Least Upper Bound, Greatest Lower Bound of a Set (cont.)

Lecture 10 - Compact Set

Lecture 11 - Properties of Compact Sets

Lecture 12 - Tutorial II

Lecture 13 - Heine-Borel Theorem

Lecture 14 - Weierstrass Theorem

Lecture 15 - Cantor Set and its Properties

Lecture 16 - Derived Set and Dense Set

Lecture 17 - Limit of a Sequence, Monotone Sequence

Lecture 18 - Tutorial III

Lecture 19 - Some Important Limits of Sequences

Lecture 20 - Ratio Test, Cauchy's Theorems on Limits of Sequences of Real Numbers

Lecture 21 - Fundamental Theorems on Limits

Lecture 22 - Some Results on Limits and Bolzano-Weierstrass Theorem

Lecture 23 - Criteria for Convergent Sequences

Lecture 24 - Tutorial IV

Lecture 25 - Criteria for Divergent Sequence

Lecture 26 - Cauchy Sequence

Lecture 27 - Cauchy Convergence Criteria for Sequences

Lecture 28 - Infinite Series of Real Numbers

Lecture 29 - Convergence Criteria for Series of Positive Real Numbers

Lecture 30 - Tutorial V

Lecture 31 - Comparison Test for Series

Lecture 32 - Absolutely and Conditionally Convergent Series

Lecture 33 - Rearrangement Theorem and Test for Convergence of Series

Lecture 34 - Ratio and Integral Test for Convergence of Series

Lecture 35 - Raabe's Test for Convergence of Series

Lecture 36 - Tutorial VI

Lecture 37 - Limit of Functions and Cluster Point

Lecture 38 - Limit of Functions (cont.)

Lecture 39 - Divergence Criteria for Limit

Lecture 40 - Various Properties of Limit of Functions

Lecture 41 - Left and Right Hand Limits for Functions

Lecture 42 - Tutorial VII

Lecture 43 - Limit of Functions at Infinity

Lecture 44 - Continuous Functions (Cauchy's Definition)

Lecture 45 - Continuous Functions (Heine's Definition)

Lecture 46 - Properties of Continuous Functions

Lecture 47 - Properties of Continuous Functions (cont.)

Lecture 48 - Tutorial VIII

Lecture 49 - Boundedness Theorem and Max-Min Theorem

Lecture 50 - Location of Root and Bolzano's Theorem

Lecture 51 - Uniform Continuity and Related Theorems

Lecture 52 - Absolute Continuity and Related Theorems

Lecture 53 - Types of Discontinuities

Lecture 54 - Tutorial IX

Lecture 55 - Types of Discontinuities (cont.)

Lecture 56 - Relation between Continuity and Compact Sets

Lecture 57 - Differentiability of Real Valued Functions

Lecture 58 - Local Max-Min Cauchy's and Lagrange's Mean Value Theorem

Lecture 59 - Rolle's Mean Value Theorems and its Applications

Lecture 60 - Tutorial X

Lecture 61 - Applications of Derivatives

Lecture 62 - Applications of Mean Value Theorem and Darboux's Theorem

Lecture 63 - L'Hospital's Rule

Lecture 64 - Taylor's Theorem

Lecture 65 - Riemann/Riemann-Stieltjes Integral

Lecture 66 - Tutorial XI

Lecture 67 - Riemann/Riemann-Stieltjes Integral (cont.)

Lecture 68 - Existence of Riemann-Stieltjes Integral

Lecture 69 - Riemann-Stieltjes Integrable Functions

Lecture 70 - Properties of Riemann-Stieltjes Integral

Lecture 71 - Various Results of Riemann-Stieltjes Integral using Step Function

Lecture 72 - Some More Results on Riemann-Stieltjes Integral

Lecture 73 - Tutorial XII