### Fall 2017

### Handouts, Quizzes, and Exams

- Syllabus
- Day 1
- How to solve problems?
- How to write correct proof by Chris Heil
- An Intuitive Development of the Real Numbers
- Quiz 1 Quiz 1 (solutions)
- Reading Comprehension
- Exam I
- The practice session for Exam I is scheduled for Monday from 4:30 to 5:30 in DMF461
- Practice Exam 1
- Practice Exam 1 (solutions)
- Exam I
- Exam I (solutions)

- Quiz 2
- Exam II
- Exam 2 (review guide)
- Exam 2
- Exam II solutions

### Flipped classroom activities

- Sections 3.1 and 3.2
- Section 3.3

### Video lectures

- Section 0
- Set and notations
- Set building notations
- Mathematical quantifiers
- Proofs with quantifiers
- Quantifiers and the empty set
- Double quantifiers
- Conditional statements
- More on conditional statements
- An example of a bad proof
- How to write a rigorous mathematical definition
- The negation of a conditional statement
- Proof by induction
- The sigma notation
- Basic proof techniques
- Introduction to the natural numbersIntroduction to the natural numbersIntroduction to the natural numbers
- Supremum and infimum
- Supremum and infimum of a function
- Denseness of the rationals
- Introduction to sequences and notations
- Examples of sequences
- The concept of convergence
- Uniqueness of a limit
- The epsilon-N approach to convergence
- An example using the epsilon-N approach
- An example using the epsilon-N approach Part 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*- Examples using the epsilon-N approach Part 3
- Examples using the epsilon-N approach Part 4

- Limits Theorem (Section 3.2)
*Errata: At 42:35 it should have been 1/|z_n|<2/|z|*- Because of the quality of the audio, please use your headphones
- Limit Theorems
- Section 3.1 and Section 3.2

- Monotone Sequences (Section 3.3)
- Because of the quality of the audio, please use your headphones
- Monotone sequences (Section 3.3)
- In-class activities for Monotone sequences (section 3.3)

- Subsequences and the Bolzano-Weierstrass Theorem
- Video lecture (Section 3.4)
- Notes for Section 3.4
- In-class activities for Subsequences and the Bolzano-Weierstrass Theorem

- Section 3.5 Cauchy Criteria
- Section 3.6 Properly Divergent Sequences
- Section 4.1 Limits of functions
- Section 4.2 Limits Theorems
- Section 5.1 Continuous functions
- Section 5.2 Combination of continuous functions
- Video lecture
- Notes
- In-class activities

- Section 5.3 Continuous functions on intervals
- Video lecture
- Notes
- In-class activities

- Section 5.4
- Video lecture
- Notes
- In-class activities

**Outline of sections and homework sets**

- Appendix A:
- A set of logic review reviewing part of MATH 180 is posted on Blackboard. Please submit your solutions for this quiz by September 13, 2017
- outline and homework
- Appendix A (solutions)

- Section 1.2 Mathematical induction
- Section 2.1 Algebraic and order properties of R
- Section 2.2 Absolute Value
- Section 2.3 The completeness property of R
- Correction of a typo: in Problem 6, you are supposed to prove that
- Homework section 2.3
- Sec 2.3 (solutions)

- Section 2.4 Application of the supremum property
- Section 2.4: Additional practice problems solutions
- Problems # 1, 3, 6, 7 (prove the result only for the case of the supremum),13,19 Section 2.4 (solutions)

- Section 3.1 Sequences and their limits
- Additional practice questions
- Problems # 3, 5, 8, 9, 11, 12, 16.
- solutions for homework

- Section 3.2 Limits theorems
- Problems # 6, 9, 13, 20.
- Section 3.2 solutions

- Section 3.3 Monotone sequences
- Problems # 4, 7, 11, 12, 14.
- Section 3.3 solutions (monotone sequences)

- Section 3.4 Subsequences and the B-W Theorem
- Problems # 1, 3, 4, 8, 9, 11.
- Section 3.4 solutions

- Section 3.5 The Cauchy Criterion
- Problems # 2, 4, 5, 7
- Solutions

- Section 3.6 Properly divergent sequences
- Problems # 4, 5, 7, 8.
- Solutions

- Section 4.1 Limits of functions
- Problems # 4,9,14
- Solutions

- Section 4.2 Limit theorems
- Problems # 1,2,4, 9, 1 0
- Solutions

- Section 5.1 Continuous functions
- Problems # 5, 6, 11, 12
- Solutions

- Section 5.2 Combination of continuous functions
- Problems # 1, 3, 12
- Solutions

- Section 5.3 Continuous functions on intervals
- Problems # 4, 6, 13
- Solutions

- Section 5.4 Uniform continuity

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