# Introductory Course in Real Analysis

- Institution: IIT Kharagpur
- Department: Mathematics
- Teacher: Prof. P D Srivastava
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111105098)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111105098)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

This is a basic course in Real Analysis which is a back bone of any course on pure & 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.

## Lectures

1. [Lecture 1: Countable and Uncountable sets](https://invisiblenetwork.ai/learn/nptel/111105098/7b70f14bdb11) (YouTube)
2. [Lecture 2: Properties of Countable and Uncountable sets](https://invisiblenetwork.ai/learn/nptel/111105098/be47b0da776b) (YouTube)
3. [Lecture 3: Examples of Countable and Uncountable sets](https://invisiblenetwork.ai/learn/nptel/111105098/01be28393be3) (YouTube)
4. [Lecture 4: Concepts of Metric Space](https://invisiblenetwork.ai/learn/nptel/111105098/e767b1aa131d) (YouTube)
5. [Lecture 5: Open ball, Closed ball, Limit point of a set](https://invisiblenetwork.ai/learn/nptel/111105098/bd297ff7e99f) (YouTube)
6. [Lecture 6: Tutorial - 1](https://invisiblenetwork.ai/learn/nptel/111105098/e1d40ed7fed4) (YouTube)
7. [Lecture 7: Some theorems on Open and Closed sets](https://invisiblenetwork.ai/learn/nptel/111105098/2e9df2220491) (YouTube)
8. [Lecture 8: Ordered set, Least upper bound, Greatest lower bound of a set](https://invisiblenetwork.ai/learn/nptel/111105098/a1314a0b6bb8) (YouTube)
9. [Lecture 9: Ordered set, Least upper bound, Greatest lower bound of a set (contd.)](https://invisiblenetwork.ai/learn/nptel/111105098/f03cd59ec922) (YouTube)
10. [Lecture 10: Compact Set](https://invisiblenetwork.ai/learn/nptel/111105098/07558f7df79f) (YouTube)
11. [Lecture 11: Properties of Compact sets](https://invisiblenetwork.ai/learn/nptel/111105098/7d3a12e65fd3) (YouTube)
12. [Lecture 12: Tutorial - 2](https://invisiblenetwork.ai/learn/nptel/111105098/ede4f05f25c9) (YouTube)
13. [Lecture 13: Heine Borel Theorem](https://invisiblenetwork.ai/learn/nptel/111105098/aceb7d14ae7d) (YouTube)
14. [Lecture 14: Weierstrass Theorem](https://invisiblenetwork.ai/learn/nptel/111105098/682a163692ff) (YouTube)
15. [Lecture 15: Cantor set & its properties](https://invisiblenetwork.ai/learn/nptel/111105098/b598cf0455ce) (YouTube)
16. [Lecture 16: Derived set & Dense set](https://invisiblenetwork.ai/learn/nptel/111105098/f14ae8d3a2eb) (YouTube)
17. [Lecture 17: Limit of a sequence & monotone sequence](https://invisiblenetwork.ai/learn/nptel/111105098/927daa1aee25) (YouTube)
18. [Lecture 18: Tutorial III](https://invisiblenetwork.ai/learn/nptel/111105098/c87f43a69ab7) (YouTube)
19. [Lecture 19 : Some Important Limits of Sequences](https://invisiblenetwork.ai/learn/nptel/111105098/8299e204b6e5) (YouTube)
20. [Lecture 20 : Ratio Test Cauchy's Theorems on Limits of Sequences of Real Numbers](https://invisiblenetwork.ai/learn/nptel/111105098/916d4d7370cd) (YouTube)
21. [Lecture 21: Fundamental Theorems on Limits](https://invisiblenetwork.ai/learn/nptel/111105098/e6d86dd52993) (YouTube)
22. [Lecture 22: Some Results on Limits and Bolzano - Weierstrass Theorem](https://invisiblenetwork.ai/learn/nptel/111105098/e99a95e15512) (YouTube)
23. [Lecture 23: Criteria for Convergent Sequence](https://invisiblenetwork.ai/learn/nptel/111105098/49ef56497c53) (YouTube)
24. [Lecture 24: Tutorial - IV](https://invisiblenetwork.ai/learn/nptel/111105098/61f3149626bf) (YouTube)
25. [Lecture 25 : Criteria for Divergent Sequence](https://invisiblenetwork.ai/learn/nptel/111105098/70744cfe3521) (YouTube)
26. [Lecture 26 : Cauchy Sequence](https://invisiblenetwork.ai/learn/nptel/111105098/657c609a03bd) (YouTube)
27. [Lecture 27 : Cauchy Convergence Criteria for Sequences](https://invisiblenetwork.ai/learn/nptel/111105098/623c1645f927) (YouTube)
28. [Lecture 28 : Infinite Series of Real Numbers](https://invisiblenetwork.ai/learn/nptel/111105098/59e515198b50) (YouTube)
29. [Lecture 29 : Convergence Criteria for Series of Positive Real Numbers](https://invisiblenetwork.ai/learn/nptel/111105098/b3aafbfb7102) (YouTube)
30. [Lecture 30 : Tutorial V](https://invisiblenetwork.ai/learn/nptel/111105098/b7266f10a65f) (YouTube)
31. [Lecture 31: Comparison Test for Series](https://invisiblenetwork.ai/learn/nptel/111105098/cc6f1d190d91) (YouTube)
32. [Lecture 32: Absolutely and Conditionally Convergent Series](https://invisiblenetwork.ai/learn/nptel/111105098/fa54a7ab86dc) (YouTube)
33. [Lecture 33: Rearrangement Theorem and Test for Convergence of Series](https://invisiblenetwork.ai/learn/nptel/111105098/c5a43032e917) (YouTube)
34. [Lecture 34: Ratio and Integral Test for Convergence of Series](https://invisiblenetwork.ai/learn/nptel/111105098/3da13b208903) (YouTube)
35. [Lecture 35: Raabe's Test for Convergence of Series](https://invisiblenetwork.ai/learn/nptel/111105098/9af01db6d997) (YouTube)
36. [Lecture 36 : Tutorial VI](https://invisiblenetwork.ai/learn/nptel/111105098/86a1cfde4f77) (YouTube)
37. [Lecture 37 : Limit of functions and cluster point](https://invisiblenetwork.ai/learn/nptel/111105098/ed61cfdcd841) (YouTube)
38. [Lecture 38 : Limit of functions(contd.)](https://invisiblenetwork.ai/learn/nptel/111105098/cdb74f702d9a) (YouTube)
39. [Lecture 39 : Divergence criteria for limit](https://invisiblenetwork.ai/learn/nptel/111105098/43094a0226ac) (YouTube)
40. [Lecture 40 : Various properties of limit of functions](https://invisiblenetwork.ai/learn/nptel/111105098/2660e779125f) (YouTube)
41. [Lecture 41 : Left and Right Hand Limits for functions](https://invisiblenetwork.ai/learn/nptel/111105098/f0eb4613b428) (YouTube)
42. [Lecture 42 : Tutorial VII](https://invisiblenetwork.ai/learn/nptel/111105098/c91d6f488376) (YouTube)
43. [Lecture 43 : Limit of Functions at Infinity](https://invisiblenetwork.ai/learn/nptel/111105098/475e77773b49) (YouTube)
44. [Lecture 44 : Continuous Functions (Cauchy's Definition)](https://invisiblenetwork.ai/learn/nptel/111105098/b586a9805bbf) (YouTube)
45. [Lecture 45: Continuous Functions (Heine's Definition)](https://invisiblenetwork.ai/learn/nptel/111105098/77a60347d7f3) (YouTube)
46. [Lecture 46: Properties of Continuous Functions](https://invisiblenetwork.ai/learn/nptel/111105098/8a1405087e58) (YouTube)
47. [Lecture 47: Properties of Continuous Functions (Contd.)](https://invisiblenetwork.ai/learn/nptel/111105098/10ee2fde2427) (YouTube)
48. [Lecture 48 : Tutorial VIII](https://invisiblenetwork.ai/learn/nptel/111105098/1c4af4367545) (YouTube)
49. [Lecture 49: Boundness Theorem and Max-Min Theorem](https://invisiblenetwork.ai/learn/nptel/111105098/991f9ce46cb1) (YouTube)
50. [Lecture 50: Location of Root and Bolzano's Theorem](https://invisiblenetwork.ai/learn/nptel/111105098/ff9b6d56face) (YouTube)
51. [Lecture 51: Uniform Continuity and Related Theorems](https://invisiblenetwork.ai/learn/nptel/111105098/e460156abf20) (YouTube)
52. [Lecture 52: Absolute Continuity and Related Theorems](https://invisiblenetwork.ai/learn/nptel/111105098/652432397d07) (YouTube)
53. [Lecture 53 : Types of Discontinuities](https://invisiblenetwork.ai/learn/nptel/111105098/7ba0711df7bb) (YouTube)
54. [Lecture 54 : Tutorial IX](https://invisiblenetwork.ai/learn/nptel/111105098/8a2ce0253ba1) (YouTube)
55. [Lecture 55 : Types of Discontinuities (Contd.)](https://invisiblenetwork.ai/learn/nptel/111105098/34dea82aeb75) (YouTube)
56. [Lecture 56 : Relation between Continuity and Compact Sets](https://invisiblenetwork.ai/learn/nptel/111105098/cb48bbe99fc5) (YouTube)
57. [Lecture 57 : Differentiability of Real Valued Functions](https://invisiblenetwork.ai/learn/nptel/111105098/8bd29b75aa28) (YouTube)
58. [Lecture 58 : Local Max. - Min. Cauchy's and Lagrange's Mean Value Theorem](https://invisiblenetwork.ai/learn/nptel/111105098/ec7811b7d936) (YouTube)
59. [Lecture 59 : Rolle's Mean Value Theorems and Its Applications](https://invisiblenetwork.ai/learn/nptel/111105098/ed4828db2c81) (YouTube)
60. [Lecture 60 : Tutorial X](https://invisiblenetwork.ai/learn/nptel/111105098/bb50e72aa82d) (YouTube)
61. [Lecture 61 : Applications of Derivatives](https://invisiblenetwork.ai/learn/nptel/111105098/384b509e6ac2) (YouTube)
62. [Lecture 62 : Application of MVT & Darboux's theorem](https://invisiblenetwork.ai/learn/nptel/111105098/7dd800be38e3) (YouTube)
63. [Lecture 63 : L' Hospital Rule](https://invisiblenetwork.ai/learn/nptel/111105098/62e78981aed1) (YouTube)
64. [Lecture 64 : Taylor's theorem](https://invisiblenetwork.ai/learn/nptel/111105098/e8dc813c1b00) (YouTube)
65. [Lecture 65 : Riemann/Riemann Steiltjes Integral](https://invisiblenetwork.ai/learn/nptel/111105098/4c68f318b746) (YouTube)
66. [Lecture 66 : Tutorial XI](https://invisiblenetwork.ai/learn/nptel/111105098/850b3ff072a0) (YouTube)
67. [Lecture 67 : Riemann/Riemann Steiltjes Integral (contd.)](https://invisiblenetwork.ai/learn/nptel/111105098/7dfb9ed2e5dd) (YouTube)
68. [Lecture 68 : Existence of Riemann/Riemann Steiltjes Integral](https://invisiblenetwork.ai/learn/nptel/111105098/8986bba391cf) (YouTube)
69. [Lecture 69 : Riemann/Riemann Steiltjes Integrable functions](https://invisiblenetwork.ai/learn/nptel/111105098/e7aa13b89fe5) (YouTube)
70. [Lecture 70 : Properties of Riemann Steiltjes Intergral](https://invisiblenetwork.ai/learn/nptel/111105098/48633f8b5498) (YouTube)
71. [Lecture 71 : Various Results of Riemann Stieltjes Integral using Step Function](https://invisiblenetwork.ai/learn/nptel/111105098/265de6448b2d) (YouTube)
72. [Lecture 72 : Some more results on Riemann Stieltjes Integral](https://invisiblenetwork.ai/learn/nptel/111105098/192617b7a5d1) (YouTube)
73. [Lecture 73 : Tutorial XII](https://invisiblenetwork.ai/learn/nptel/111105098/298fa6bbddcb) (YouTube)