A Basic Course in Real Analysis
A Basic Course in Real Analysis. Instructor: Prof. P.D. Srivastava, Department of Mathematics, IIT Kharagpur. This is an introductory course in real analysis, covering topics: the Dedekind theory of irrational numbers; the Cantor theory of irrational numbers; bounded set and the open, closed and compact, etcetera; continuity of the function; differentiability; the Riemann integration and Riemann stieltjes integral; and the improper integral.
(from nptel.ac.in )

Lecture 12 - Compact Sets and its Properties
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Lecture 01 - Rational Numbers and Rational Cuts
Lecture 02 - Irrational Numbers, Dedekind's Theorem
Lecture 03 - Continuum and Exercises
Lecture 04 - Continuum and Exercises (cont.)
Lecture 05 - Cantor's Theory of Irrational Numbers
Lecture 06 - Cantor's Theory of Irrational Numbers (cont.)
Lecture 07 - Equivalence of Dedekind and Cantor's Theory
Lecture 08 - Finite, Infinite, Countable and Uncountable Sets of Real Numbers
Lecture 09 - Types of Sets with Examples, Metric Space
Lecture 10 - Various Properties of Open Set, Closure of a Set
Lecture 11 - Ordered Set, Least Upper Bound, Greatest Lower Bound of a Set
Lecture 12 - Compact Sets and its Properties
Lecture 13 - Weierstrass Theorem, Heine Borel Theorem, Connected Set
Lecture 14 - Tutorial II
Lecture 15 - Concept of Limit of a Sequence
Lecture 16 - Some Important Limits, Ratio Tests for Sequences of Real Numbers
Lecture 17 - Cauchy Theorems on Limit of Sequences with Examples
Lecture 18 - Fundamental Theorem on Limits, Bolzano-Weierstrass Theorem
Lecture 19 - Theorems on Convergent and Divergent Sequences
Lecture 20 - Cauchy Sequence and its Properties
Lecture 21 - Infinite Series of Real Numbers
Lecture 22 - Comparison Tests for Series, Absolutely Convergent and Conditional Convergent Series
Lecture 23 - Tests for Absolutely Convergent Series
Lecture 24 - Raabe's Test, Limit on Functions, Cluster Point
Lecture 25 - Some Results on Limit of Functions
Lecture 26 - Limit Theorems for Functions
Lecture 27 - Extension of Limit Concept (One Sided Limits)
Lecture 28 - Continuity of Functions
Lecture 29 - Properties of Continuous Functions
Lecture 30 - Boundedness Theorem, Max-Min Theorem, and Bolzano's Theorem
Lecture 31 - Uniform Continuity and Absolute Continuity
Lecture 32 - Types of Discontinuities, Continuity and Compactness
Lecture 33 - Continuity and Compactness (cont.), Connectedness
Lecture 34 - Differentiability of Real Valued Function, Mean Value Theorem
Lecture 35 - Mean Value Theorem (cont.)
Lecture 36 - Application of Mean Value Theorem, Darboux Theorem, L'Hospital's Rule
Lecture 37 - L'Hospital's Rule and Taylor's Theorem
Lecture 38 - Tutorial III
Lecture 39 - Riemann/Riemann-Stieltjes Integral
Lecture 40 - Existence of Riemann-Stieltjes Integral
Lecture 41 - Properties of Riemann-Stieltjes Integral
Lecture 42 - Properties of Riemann-Stieltjes Integral (cont.)
Lecture 43 - Definite and Indefinite Integral
Lecture 44 - Fundamental Theorems of Integral Calculus
Lecture 45 - Improper Integrals
Lecture 46 - Convergence Test for Improper Integrals