Math 112 Real Analysis



Welcome to Math 112 Real Analysis! In this course, you will learn to admire the formal definition of the limit of a function (and much more), just like our friends and definers of the limit, Bernard Bolzano and Karl Weierstrass.

Class meets in Science Center Hall E on MWF, 1-2pm.

Note: Every semester, I try to pick up one new html trick to incorporate into my website. This semester's trick is collapsable lists! Clicking on the headers below will expand the subject, often revealing new collapsable lists. This allows me to include a lot of detail and justification on course policies, etc., without cluttering everything to the point of unreadability. The announcements, reading assignments, and problem sets are always displayed, but I encourage you to explore the rest of the material as well.

Announcements:

Our CA Emily is typing up notes as the course goes.

Notes


  • The course policies are described throughout this webpage. They are also outlined in the syllabus.
  • The most important thing is that you respect your fellow classmates and yourselves in all interactions. We've all taken different paths to get here, but we've come together for the shared goal of learning real analysis.
  • Homework: 50%
  • High Midterm score: 20%
  • Low Midterm score: 10%
  • Final: 20%



  • Email: wboney@math.harvard.edu
  • Office: Science Center 238

    My office is kind of hard to find the first time, so here's a map (follow the red path, it starts at the second floor elevators). If you come out of the main elevators on the second floor, take a u-turn to your left and go all the way to the end of the hallway. Go through the door at the end to enter a cluster of math department offices. If you keep walking back in that direction, you'll find my office.
  • Office hours:
    • Tuesday 4-5pm
    • Wednesday 10-11am
    • Friday 12-1pm
    Office hours are a great time to come and ask questions about course material and/or homework. Note that my office hours are not right before homework is due. This is by design, to encourage you to start looking at the homework early.
    Email: esaunders@college.harvard.edu
  • Office hours: M 6-7, SC 222
  • Section: F 3-4, SC 307
    Email: salashnabaala@college.harvard.edu
  • Office Hours: M 8-9, Leverett Dining Hall (Math Night)
  • Section: Sun 5-6, SC 110 and T 1-2, SC 222
    Email: g_whitney@college.harvard.edu
  • Office Hours: T 3-4, SC 113

Assignment .pdf .tex Due Date
Week 1 PS1 PS1 Jan. 31
I removed some of the problems, but they will appear on a later problem set.
Week 2 PS2 PS2 Feb. 7
Week 3 PS3 PS3 Feb. 14
Week 4 Cancelled due to midterm.
Week 5 PS5 PS5 Feb. 28
Week 6 PS6 PS6 March 9
Since the following week is Spring Break, there are some changes to this assignment: due on a Friday, no Doc Brown problems, and includes material from the week it's due in
Week 7 PS7 PS7 March 28
Week 8 PS8 PS8 April 4
Week 9 Cancelled due to midterm.
Week 10 PS10 PS10 April 18
Week 11 PS11 PS11 April 25


  • Actively engaging with the material is the best way to learn math. I want to give you as much opportunity to do so as possible, so we will have weekly problem sets due Wednesday. In particular, you can turn in problem sets either in class or via email to the CAs (please include all three on the email). Either way, the problem sets are due by the end of class.
  • Math is a team sport and the best way to learn is to find classmates and friends to work on problem sets with. I really encourage you to work with people in this class both on completing the problem sets and gaining a better understanding of the material in general.
  • Math Night is a great place to meet up with people in your class and in the larger department. Note that the homework is cleverly scheduled to be due the class after Math Night each week.
  • To really drive home that I'm serious about collaboration and to help you find people to work with, for the first three assignments, I will be assigning you groups to work with. I'll send out emails to groups around the same time that I post assignments. These aren't the only people you can work with, but I'd like you to try working in these groups for at least an hour. If you do, then your group will get 10 bonus points on the assignment. Make sure everyone writes at the top of their homework that y'all worked together so the grader's know.

    If you run into problems meeting up with your group (some group members don't want to meet, but others do; there's no time that works for everyone; etc.), let me know and I'll help resolve the problem.
  • Doc Brown problems are problems from the future, designed to get you thinking about the material we will cover the coming week. These will be simpler in nature and should be doable from the readings.
  • Expectations: You are expected to turn in all of the Core and Doc Brown problems each week. However, each Niche problem can replace a Core problem (sometimes more, this is indicated on the problem sheets). Additionally, throughout the semester, you are required to complete at least 15 10 Niche problems total (that's a little more than one a week).

    In a world where Real Analysis was your life, I'd have you do all of all of the problems. Recognizing that you have other demands on your time, I don't require that you do all Niche problems every week. However, I encourage you to at least think about how you would solve all of the problems whenever possible.
  • Example: Homework 1 has 8 Core problems, 3 Doc Brown problems, and 2 Niche problems. You could choose to complete all Core and Doc Brown problems, and no Niche problems.. Alternatively, you could complete 6 Core problems, 3 Doc Brown problems, and 2 Niche problems.
  • LaTeX or TeX is a typesetting program that allows you to easily type the mathematical symbols that you'll need to use on your homework.
  • Here is a template on Overleaf that will let you dive right in. There will be a LaTeX how-to seminar hosted by our course assistant.

    Another useful tool is DeTeXify is another useful tool that allows you to draw the symbol you want, and it will help you find the right code. I often use the DeTeXify app on my phone when trying to use new symbols.
  • TeXing your homework is a valuable skill to learn in math, and helps greatly streamline the process of grading and resubmitting. In this class, there's a scaling requirement of TeXing your homework. For Week 1 problem set, there's no TeXing requirement. Each week, the number of required problems that are required to be TeXed increases by 1 until Spring Break. After Spring Break, you are required to TeX your entire problem set.
  • We all make mistakes, and the best thing to do is to learn from these mistakes. Thus, if you get back a problem set that you wish you had done better on, you can resubmit up to one week after the problem sets are returned. You're eligible for up to half of the points missed. Also, to avoid gaming of this combined with no late homework, you can only resubmit problems that you actually submitted in the first place.
  • The homework is designed to help you check, solidify, and deepen your understanding of the material. Since this course builds on itself (as most do), this is less helpful if you delay it. Thus, late homework is not accepted.
  • Homework should be turned in by the end of class, either a hard copy or an email.
  • I understand that things come up, and so you are given two 'grace' assignments throughout the semester. These are assignments that you can turn in up to one week late. If something comes up outside of the scope of these two grace assignments, please discuss it with me individually.

Week Reading from Marsden and Hoffman
Week 0 Introduction ("On the History of Set Theory" and "The Axioms" are not necessary)
Week 1 1.1-1.3
I would like you review the Week 0 reading before the first day of class; it covers some basic concepts and logic. Then complete the rest of the Week 1 reading as we progress through the week. Remember that many theorem proofs are delayed until the end of the chapter!
Week 2 1.3-1.4, (1.5, 1.6, 1.7)
We won't be covering the later chapter this week, but you should still take a brief look at them to do the Doc Brown problems on the HW. As a hint, two of them just requiring plugging things into a formula!
Week 3 1.5-1.6, 2.1-2.3
In reading Chapter 2 and beyond, don't worry about thinking about general metric spaces. It's enough to think of R^n
Week 4 2.1-2.8
Week 5 2.9-3.3
Week 6 Happy Spring Break
Week 7 4.1-4.5


  • It's important to do the reading for class. This will give you some initial exposure to the material prior to coming to class.
  • Come to class with questions from the reading! I don't expect you to obtain a perfect understanding of the material from reading, and you should use class time as an opportunity to clear up any confusion from the reading.
  • You should aim to complete the reading prior to the start of the week. Typically, we class will progress through the text as written, but sometimes things will be covered in a different order. Additionally, the Doc Brown problems are written with this schedule in mind.
  • Keep in mind that you will be responsible both for the material covered in class and in the readings!

  • The midterms will be taken home exams. They will be avaialable over a weekend and you are allowed to work on them during a (contiguous) three hour window of your choice. Choose your time and location carefully because there is no 'pausing' of the time. You should record your starting and ending time on the exam, and return the completed exam to my office by noon on the following Monday.

    Late exams will not be accepted. YOu can also email your exam to me, but all work including TeXing must be complete in the three hour window.

    You are not permitted to discuss the exam with anyone until all exams have been turned in. You are also not permitted to consult any resources (books, notes, the internet, etc.) except yourself and a single page of notes as outlined below.

    The only outside resource allowed is a single side of an 8 1/2 by 11 piece of paper of notes on the exam. This sheet must be prepared prior to the exam period begining and must be emailed to me before the start of the exam window. You should use a hard copy of the sheet and not use any sort of magnification to help you read.
  • The first midterm will probably occur over the weekend Feb 16-19 (President's Day weekend) and cover Ch 1-3. This will mainly serve as a diagnostic tool to see how the class is progressing.
  • The second midterm will probably occur over the weekend Apr 6-8 and cover Ch 4-6 (and the previous material).
  • We will have a three-hour in class final exam during the time scheduled by the registrar. It will be comprehensive.
  • Our final is scheduled for May 3, 9:00am.

There will be two (optional) workshops run by the CA: one on writing proofs and one on using TeX
  • Time: TBD
  • Location: TBD
  • Time: TBD
  • Location: TBD

  • I'm happy to accommodate any modifications suggested by the Accessible Education Office (AEO). Please be sure to provide me with the introductory letter as soon as possible.
  • I expect students at all time to follow Harvard's Honor Code and practice academic integrity. I understand that the line between collaboration and plagiarism can be murky on problem sets, but the following is a good (and oft quoted) rule of thumb: working together is fantastic, but you should always be able to separate and right up your solutions individually. If you have any questions, please don't hesitate to ask.

If you have any questions, please feel free to contact me at wboney@math.harvard.edu.
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