# Real Analysis

## Spring 2018

Study-guide for the final exam

Our final exam will cover the following topics:

• Appendix A Logic and Proofs
• 2 Mathematical induction
• 2.1 The Algebraic and order properties of R
• 2.3 The completeness property of R
• 2.4 Application of the supremum property
• 3.1 Sequences and their limits
• 3.2 Limits theorems
• 3.3 Monotone sequences
• 3.4 Subsequences and the B-W Theorem
• 3.5 The Cauchy Criterion
• 3.6 Properly divergent sequences
• 4.1 Limits of functions
• 4.2 Limit theorems
• 5.1 Continuous functions
• 5.2 Combination of continuous functions
• 5.3 Continuous functions on intervals (whatever was covered in class)

As a strategy going into the final, I recommend that you know the definitions of all key terms and that you are able to faithfully recall, and state all main theorems covered during the course of the semester. Your pool of questions consists of all in-class activities, the three in-class exams as well as the quizzes. I have posted on our blog all relevant materials for you to study. You will also find the blog complete solutions to quizzes and exams. As such, I would say that the materials organized under Spring 2018 on https://vignonoussa.wordpress.com/real-analysis/ are a comprehensive and complete study-guide. I am reminding you that studying for a final exam for this type of course is (and should be) time-consuming. Your review should be exhaustive. However, you should also pace yourself so that the task does not feel so daunting and overwhelming. Perhaps it would be wise to study a couple hours every day until the exam.

The format of the exam will be like that of the in-class exams. The first part of the exam will consist of recalling definitions, stating theorems and giving examples. Although, the first part is usually straightforward, please, do not be casual about your answers here. You really want to be precise and thorough. You should take advantage and maximize your performance in the first part of the exam since your ability to do well on Part 1 mainly relies on memorization and the construction of correct examples.  The second part of the exam will contain a list of statements to prove. You will be given the freedom to select a certain number of problems to solve. I repeat here that these questions will be mainly inspired by the in-class activities, exam questions, and quizzes. As you are reviewing keep in mind that just memorizing solutions of proofs is not a good strategy. It is nearly impossible to write a correct proof without having a clear and complete understanding of the fundamental and core ideas. I recommend that you take time to also understand the structure of the proofs and that you pay attention to the coherency and logical steps taken in each proof.

Quizzes and Exams

### 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|
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