Spring 2018
Studyguide 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 BW 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 inclass activities, the three inclass 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/realanalysis/ are a comprehensive and complete studyguide. I am reminding you that studying for a final exam for this type of course is (and should be) timeconsuming. 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 inclass 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 inclass 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
 Appendix A
 Video Lecture
 Correction by Zenan: At 8:40, the result in the bracket should be [(2l)^3+3(2l)^2+3(2l)+1].
However, it doesn’t affect the answer at all.
 Correction by Zenan: At 8:40, the result in the bracket should be [(2l)^3+3(2l)^2+3(2l)+1].
 Appendix
 In class activities
 Outline and homework and Appendix (solutions)
 Writing Proofs by Chris Heil
 Video Lecture
 Section 1.2 Mathematical Induction
 Video Lecture
 Inclass activities
 How to write a correct proof by induction
 Inclass Activities
 Homework and Solutions
 Section 2.1 The Algebraic and Ordering Property of the Reals
 Section 2.2 Absolute value and the real line
 Section 2.3 The Completeness property of R
 Video Lecture
 Inclass Activities
 Homework section 2.3 Sec 2.3 (solutions)
 Correction of a typo: in Problem 6, you are supposed to prove that
 Section 2.4 Applications of the supremum property
 Video Lecture
 Inclass Activities
 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 limits
 Video Lectures
 Links: the length of the lectures is in total about 90 min

 In class activities Section 3.1 and Section 3.2
 Additional practice questions
 Problems # 3, 5, 8, 9, 11, 12, 16.
 solutions for homework
 Video Lectures
 Section 3.2 Limit theorems
 Video Lecture Errata: At 42:35 it should have been 1/z_n<2/z, Because of the quality of the audio, please use your headphones
 Lecture notes Limit Theorems
 In class activities Section 3.1 and Section 3.2
 Problems # 6, 9, 13, 20. Section 3.2 solutions
 Section 3.3 Monotone sequences
 Video Lectures Monotone Sequences (because of the quality of the audio, please use your headphones)
 Lecture notes
 Inclass activities for Monotone sequences (section 3.3)
 Problems # 4, 7, 11, 12, 14. Section 3.3 solutions (monotone sequences)
 Section 3.4 Subsequences and the BolzanoWeierstrass Theorem
 Video lecture (Section 3.4)
 Notes for Section 3.4
 Inclass activities
 Problems # 1, 3, 4, 8, 9, 11. Section 3.4 solutions
 Section 3.5 Cauchy Criteria
 Video Lecture (Section 3.5)
 Notes Cauchy sequences
 Inclass activities
 Problems # 2, 4, 5, 7 Solutions
 Section 3.6 Properly Divergent Sequences
 Video Lecture (section 3.6)
 Notes for Section 3.6
 Inclass activities
 Problems # 4, 5, 7, 8. Solutions
 Section 4.1 Limits of functions
 Video Lecture Part I
 Video Lecture Part II
 Notes (handwritten)
 Notes (typed version)
 Inclass activities
 Problems # 4,9,14 Solutions
 Section 4.2 Limits Theorems
 Video Lecture
 Notes
 Notes (typed up version)
 Inclass activities
 Problems # 1,2,4, 9, 1 0 Solutions
 Section 5.1 Continuous functions
 Video lecture
 Notes
 Typed up notes
 Inclass activities
 Problems # 5, 6, 11, 12 Solutions
 Section 5.2 Combination of continuous functions
 Video lecture
 Notes
 Inclass activities
 Problems # 1, 3, 12 Solutions
 In class notes Section 5.2
 Section 5.3 Continuous functions on intervals
 Video lecture
 Notes
 Inclass activities
 Problems # 4, 6, 13 Solutions
 Section 5.4
 Video lecture
 Notes
 Inclass activities
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
 Quiz 2
 Quiz2solutions
 (Download) Solutions in video format
 Exam II
 Exam 2 (review guide)
 Exam 2
 Exam 2 solutions
 Quiz 3
 Quiz 3
 Quiz 3 (solutions)
 Exam III
 Exam 3
 Exam 3 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 epsilonN approach to convergence
 An example using the epsilonN approach
 An example using the epsilonN 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 epsilonN approach Part 3
 Examples using the epsilonN 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)
 Inclass activities for Monotone sequences (section 3.3)
 Subsequences and the BolzanoWeierstrass Theorem
 Video lecture (Section 3.4)
 Notes for Section 3.4
 Inclass activities for Subsequences and the BolzanoWeierstrass 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
 Section 5.3 Continuous functions on intervals
 Section 5.4
 Video lecture
 Notes
 Inclass 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 BW 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