Math 117: Real Analysis

Professor: Katy Craig, katy•craig at ucsb • edu , SH 6507

Teaching Assistant: Jose Zavala-Fonseca, jtz at ucsb • edu

Syllabus:

Flipped Class: This is a flipped class. Before class, you will watch lectures asynchronously. During class, you will work in groups on the homework, and I will give targeted "mini-lectures" to explain difficult points.

Office Hours:
Craig: Thursday 2:30-3:30pm and Friday 1-2pm in SH 6507 (Starting 5/2 office hours will be Monday 2-3pm instead of Thursday)
Zavala-Fonseca: Thursday 5-7pm in math lab

Grading Scheme: homework: 20%, quizzes 15%, highest midterm: 30%; final: 35%
If you have questions about the grading of any assignment or exam, you have one week after it is graded to request a regrade via gradescope.

Prerequisites: Math 8

Textbook: Elementary Analysis by Kenneth Ross, 2nd edition
Using the above link, you can purchase a paperback copy for $24.99 and download a PDF version for free. Do both.

Discord Server: We have a class discord server, where you can ask questions about the course and collaborate with your classmates. A link to the server is available at the top of the Gauchospace page. Please treat this as a public channel and do not post private information.

Quizzes:
  • Each class, there will be an open note, multiple choice quiz that asks questions about the video lectures assigned for that day.
  • Quizzes will be administered using gradescope, and students can take the quiz either on their phone or computer, with paper as a backup if there are technological difficulties.
  • Quiz questions will be extremely straightforward if you have watched the lecture and taken notes.
  • There will be no retaking of quizzes under any circumstances. Since I understand that unexpected things happen, I will automatically drop your four lowest quiz scores.

Exams:
  • There will two in class midterms and one in class final exam.
  • In order to remove as much time pressure as possible, you will be given twice the amount time I have given in the past for the same length of exam. In particular, midterms will take place over two days, with half the exam being given on each day.
  • There will be no retaking or rescheduling of exams under any circumstances. Since I understand that unexpected things happen, I will automatically drop whichever score is lowest: Midterm 1 or Midterm 2.

    • Midterm 1a: Tuesday, April 19th
    • Midterm 1b: Thursday, April 21st
    • Midterm 2a Tuesday, May 17th
    • Midterm 2b: Thursday, May 19th
    • Final Exam: Thursday, June 9th, 4-7pm
Homework:
  • Homework assignments will be posted on the course website and will be due Mondays at 11:59pm.
  • Homework will be turned in via gradescope.
  • Only problems marked with an asterisk (*) should be submitted for grading.
  • At least one problem on each of the exams will be chosen from the non-asterisked homework problems.
  • No late homework will be accepted.
  • The TWO lowest homework grades will be dropped and will not count toward the final grade.



Weekly Routine:

Monday Tuesday Wednesday Thursday Friday
  • watch assigned videos and take notes for quiz
  • submit problem set by 11:59pm.
  • attend class
  • watch assigned videos and take notes for quiz
  • attend class
  • finish problem set


Outline of Course:

Part I: Sequences Part II: Functions
the real numbers, inf, and sup continuous functions
limit, liminf, limsup cts functions attain max and min on closed interval
bounded, monotone, and Cauchy sequences intermediate value theorem
subsequences and the Bolzano-Weierstrass theorem uniform continuity and limits of functions


Daily Course Materials: (updated throughout quarter)

week day video reading/study materials due today
1 3/29 (T) [in class lecture on N,Z,Q,R & induction] LEC1
Sec1-2, appendix
1 3/31 (Th) VID1c: absolute value
VID1d: sqrt(2) is not rational
VID2a: fields
VID2b: ordered fields
VID2c: supremum, infimum, defn of R
LEC2
Sec3
2 4/5 (T) VID3a: supremum and infimum, again
VID3b: Archimedean property
VID3c: Q is dense in R
LEC3
Sec4-5
HW1, HW1Sol
2 4/7 (Th) VID4a:sequences
VID4b:convergent/divergence sequences
VID4c:bounded sequences
Warning: there is a small type-o at the end of VID4c. The floor function should be defined as ⌊N⌋= max{n : n ∈ \mathbb{N} s.t. n ≤ N}. (I accidentally wrote "min" instead of "max".)
LEC4
Sec7-8
3 4/12 (T) VID5a: limit of sum
VID5b: limit of product
VID5c: examples, divergence to infinity
LEC5
Sec9
HW2
HW2Sol
3 4/14 (Th) VID6a: bounded monotone sequences converge
VID6b: general monotone sequences
LEC6
Sec10
4 4/19 (T) Midterm 1a covering lectures 1-5 PracMid1
PracMid1SOL
Mid1a
4 4/21 (Th) Midterm 1b covering lectures 1-5 Mid1b
Mid1SOL
5 4/26 (T) VID7a: limsup and liminf
VID7b: when limsup = liminf
LEC7
Sec10
5 4/28 (Th) VID8a: Cauchy sequences
VID8b: Cauchy iff convergent
Warning: In Lecture 8a, there is a small type-o in the proof that Cauchy sequences are bounded, in which \epsilon = 3 becomes \epsilon = 2.
LEC8
Sec10
6 5/3 (T) VID9a: subsequences
VID9b: subsequential limits
Warning: There is a small type-o in the proof n_k \geq k, where I write > instead of \geq. Please see the pdf where this type-o is corrected.
LEC9
Sec11
HW3
HW3SOL
HW3Q5SOL (thanks Sam Garcia!)
6 5/5 (Th) VID10a: bounded seq. have convergent subseq.
VID10b: subsequences, liminf, and limsup
LEC10
Sec12
7 5/10 (T) VID11a: sequences and series
VID11b: continuous functions
VID11c: example of epsilon/delta defn
Warning: there is a type-o in the proof of the comparison theorem, where t_n should be defined as the sum up to n, not the sum up to infinity.
LEC11
Sec14 and Sec 17
HW4
HW4SOL
7 5/12 (Th) VID12a: more continuous functions
VID12b: combining continuous functions
LEC12
Sec17
8 5/17 (T) Midterm 2a covering lectures 1-11 PracMid2
PracMid2SOL
Midterm 2a
8 5/19 (Th) Midterm 2b covering lectures 1-11 Midterm 2b
Mid2SOL
9 5/24 (T) VID13a: attaining maximum/minimum
VID13b: intermediate value theorem
LEC13
Sec18
9 5/26 (Th) VID14a: uniform continuity, part one
VID14b: uniform continuity, part two
VID14c: uniform continuity, part three
VID14d: cts on [a,b] implies unif cts
VID14e: unif cts sends cnvgt to cnvgt
LEC14
Sec19
OfficeHours_052722
10 5/31 (T) VID15a: limits of functions, part one
VID15b: limits of functions, part two
LEC15
Sec20
HW5 HW5SOL
10 6/2 (Th) VID16a: limits of functions
VID18b: review of triangle inequality
VID18c: review of Archimedean property, part one
VID18d: review of Archimedean property, part two
VID18e: review of subsequences
VID18f: review of continuous functions
LEC18 OfficeHours_060222
- 6/9 (Th) Final Exam covering lectures 1-18 , ZoomLink

You must log in to zoom with your video on while taking the final exam on Gradescope.
PracticeFinal PracticeFinalSOL
FinalExam


Extra Credit Math Movie Competition:
As an opportunity for extra credit, we will hold a math movie competition. The goal is to make the best math movie, lasting three minutes or less. Submissions are due at 11:59pm on Sunday, May 22nd. The winner of the competition will receive ten points of extra credit on their final exam. Second place will receive five points of extra credit, and third place will receive three points of extra credit. (You are allowed to work in groups, but then the extra credit points will be distributed equally among all members of the group.)

Submissions should be uploaded to YouTube, Vimeo, or a similar site. Links to the movies can be emailed to me. (Please do not send the movies as email attachments.)

Potential topic ideas for inspiration...

Do's and Don'ts:

  • Do let me know if you choose one of the above topics, so I can remove it from the list, to prevent duplicates.
  • Do use your video as a chance to feature yourself, your roommates, your drawings... anything you create!
  • Do show a list of references at the end of the video, including any articles, books, or websites you consulted while making the video.
  • Do not simply use clunky online tools to quickly make a cartoon. I get tons of these every year, and I have yet to see one that displays creativity.
  • Do not plagiarize. Some students have simply made a video of themselves reading something they found on the internet, without attribution. This is bad.

Here are some of my favorite videos from previous years: