# Real Analysis

- Institution: Massachusetts Institute of Technology
- Department: Mathematics
- Teacher: Dr. Casey Rodriguez
- Term: Fall 2020
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [MIT OpenCourseWare](https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/)
- Licence: [CC BY-NC-SA 4.0](https://creativecommons.org/licenses/by-nc-sa/4.0/) (Non-commercial use only.)
- Syllabus: [at the source](https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/pages/syllabus/)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts through a study of real numbers, and teaches an understanding and construction of proofs.

## Lectures

1. [Lecture 1: Sets, Set Operations and Mathematical Induction](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/4d73c0245288) (Video)
2. [Lecture 2: Cantor's Theory of Cardinality (Size)](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/47c2e3aa728d) (Video)
3. [Lecture 3: Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/0fad03dc7202) (Video)
4. [Lecture 4: The Characterization of the Real Numbers](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/19b560b83981) (Video)
5. [Lecture 5: The Archimedian Property, Density of the Rationals, and Absolute Value](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/9c42d062bd0e) (Video)
6. [Lecture 6: The Uncountabality of the Real Numbers](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/d3218a2aae27) (Video)
7. [Lecture 7: Convergent Sequences of Real Numbers](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/8855f490026c) (Video)
8. [Lecture 8: The Squeeze Theorem and Operations Involving Convergent Sequences](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/5878d0fde76d) (Video)
9. [Lecture 9: Limsup, Liminf, and the Bolzano-Weierstrass Theorem](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/f8392e5b518b) (Video)
10. [Lecture 10: The Completeness of the Real Numbers and Basic Properties of Infinite Series](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/2afdf88f8c09) (Video)
11. [Lecture 11: Absolute Convergence and the Comparison Test for Series](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/9c245368b910) (Video)
12. [Lecture 12: The Ratio, Root, and Alternating Series Tests](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/c65f7fcc372a) (Video)
13. [Lecture 13: Limits of Functions](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/06574b37e3eb) (Video)
14. [Lecture 14: Limits of Functions in Terms of Sequences and Continuity](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/2028a2ac06c0) (Video)
15. [Lecture 15: The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet's Function](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/8a2fc72b2584) (Video)
16. [Lecture 16: The Min/Max Theorem and Bolzano's Intermediate Value Theorem](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/ae466413044a) (Video)
17. [Lecture 17: Uniform Continuity and the Definition of the Derivative](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/851ce3c45a58) (Video)
18. [Lecture 18: Weierstrass's Example of a Continuous and Nowhere Differentiable Function](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/8f96e0fd4a28) (Video)
19. [Lecture 19: Differentiation Rules, Rolle's Theorem, and the Mean Value Theorem](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/0c94ea6820e5) (Video)
20. [Lecture 20: Taylor's Theorem and the Definition of Riemann Sums](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/090ff1961aea) (Video)
21. [Lecture 21: The Riemann Integral of a Continuous Function](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/f3456f90562e) (Video)
22. [Lecture 22: Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/e70363e42b4d) (Video)
23. [Lecture 23: Pointwise and Uniform Convergence of Sequences of Functions](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/5212532164ca) (Video)
24. [Lecture 24: Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/2fb07036bfd4) (Video)
25. [Lecture 25: Power Series and the Weierstrass Approximation Theorem](https://invisiblenetwork.ai/learn/mit/18-100a-real-analysis-fall-2020/852e5247ff5c) (Video)