# Real Analysis

## Spring 2018

### Flipped classroom activities

• Sections 3.1 and 3.2
• Section 3.3

### Video lectures

1. Section 0
2. Set and notations
3. Set building notations
4. Mathematical quantifiers
5. Proofs with quantifiers
6. Quantifiers and the empty set
7. Double quantifiers
8. Conditional statements
9. More on conditional statements
10. An example of a bad proof
11. How to write a rigorous mathematical definition
12. The negation of a conditional statement
13. Proof by induction
14. The sigma notation
15. Basic proof techniques
16. Introduction to the natural numbersIntroduction to the natural numbersIntroduction to the natural numbers
17. Supremum and infimum
18. Supremum and infimum of a function
19. Denseness of the rationals
20. Introduction to sequences and notations
21. Examples of sequences
22. The concept of convergence
23. Uniqueness of a limit
24. The epsilon-N approach to convergence
25. An example using the epsilon-N approach
1. An example using the epsilon-N approach Part 2
2. Errata: In the part of the formal proof, toward the end, I should not have written for all n in the natural number right before the conclusion
3. Examples using the epsilon-N approach Part 3
4. Examples using the epsilon-N approach Part 4
26. Limits Theorem (Section 3.2)
1. Errata: At 42:35 it should have been 1/|z_n|<2/|z|
2. Because of the quality of the audio, please use your headphones
3. Limit Theorems
4. Section 3.1 and Section 3.2
27. Monotone Sequences (Section 3.3)
28. Subsequences and the Bolzano-Weierstrass Theorem
29. Section 3.5 Cauchy Criteria
30. Section 3.6 Properly Divergent Sequences
31. Section 4.1 Limits of functions
32. Section 4.2 Limits Theorems
33. Section 5.1 Continuous functions
34. Section 5.2 Combination of continuous functions
35. Section 5.3 Continuous functions on intervals
36. Section 5.4
1. Video lecture
2. Notes
3. In-class activities

Outline of sections and homework sets