Instructor: Tim Chumley
Office: Clapp 423
Phone: 538-2525
e-mail: tchumley
Office Hours: tentatively Mondays & Wednesdays 4:00-5:00, Thursdays 1:00-2:00; additional availability by appointment
Textbook: Understanding Analysis by Stephen Abbott, ISBN: 9781493927111;
available as a free e-text.
Announcements
Announcements will be posted on the Course Announcements Moodle forum throughout the semester, but essentially all other materials will be posted on this page.
Syllabus
Check the syllabus for all the important class policies (grades, attendance, etc.).
Homework
There will be weekly homework assignments throughout the semester to be turned in. Please read these guidelines for writing mathematics.
- General information. A selection of problems will be assigned to be written up and turned in each week.
- These problems will be due Fridays at 5 pm.
- You may work with others but the writing should be done on your own unless otherwise specified.
- Gradescope. Homework will be turned in through Gradescope.
- You should be enrolled automatically if you have an account with your MHC email address. Please let me know if you have any issues logging in.
- Gradescope has made a short tutorial on submitting homework.
- Collaboration. I want you to work together on the homework! The process of explaining your ideas to one another really helps in learning math. However, you must write up the assignments on your own and avoid copying others’ work directly. Also, please only write what you understand so that I know where to help, and please avoid using online forums like Math StackExchange, solutions manuals, or similar resources. A huge part of learning in this class comes from figuring out how to get unstuck by spending time thinking on your own or talking to me and classmates; these other resources can end up being counter-productive in the long term.
- Rewrites. Homework is for practice, and you are not expected to write perfect answers from the start!
- You will be allowed to submit revisions of any problems for full credit each week.
- Your revisions will be due on Fridays at 5 pm. This means each week you’ll have two things to turn in on Fridays: an initial submission and a redo submission.
- Please resubmit (only the problems you’re revising) on Gradescope by using the resubmit function. I’ll be able to see your submission history in order to see what was initially correct or incorrect.
| Assignment | Due |
|---|---|
| Homework 0 | Jan 31 |
| Homework 1 | Feb 7 |
| Homework 2 | Feb 14 |
| Homework 3 | Feb 21 |
| Homework 4 | Feb 28 |
| Homework 5 | Mar 14 |
| Homework 6 | Mar 28 |
| Homework 7 | Apr 4 |
| Homework 8 | Apr 18 |
| Homework 9 | Apr 25 |
| Homework 10 | May 2 |
Quizzes
There will be quizzes most weeks that will be given on Fridays at the end of class. The purpose of these is to check in to see that you’re comfortable with fundamental material and homework problems. Problems will always be related to the previous homework and class topics.
| Quiz | Date | Material |
|---|---|---|
| Quiz 1 | Feb 7 | Homework 0 |
| Quiz 2 | Feb 14 | Homework 1 |
| Quiz 3 | Feb 21 | Homework 2 |
| Quiz 4 | Feb 28 | Homework 3 |
| Quiz 5 | Mar 14 | TBA |
| Quiz 6 | Mar 28 | Homework 5 |
| Quiz 7 | Apr 4 | Homework 6 |
| Quiz 8 | Apr 18 | TBA |
| Quiz 9 | Apr 25 | Homework 8 |
| Quiz 10 | May 2 | Homework 9 |
Exams
There will be three exams. The dates for the exams are subject to change slightly.
| Exam | Due | Format | Material |
|---|---|---|---|
| Exam 1 | Mar 7 | TBA | Homework 0-4 |
| Exam 2 | Apr 11 | TBA | Homework 5-7 |
| Exam 3 | May 9-13 | self-scheduled | Homework 8-10 |
Course plan
Below is a rough outline of what is to be covered week by week through the semester. Please check back regularly for precise details on what is covered, as well as postings of class materials like lecture notes. A note about files: LaTeX and LyX source code is available for nearly all typeset documents. Simply change the file extension from .pdf to .tex or .lyx in the URL.
Chapter 1
Wednesday
- Topic: Section 1.1: Introduction. We discuss general aims for the study of real analysis; namely, we hope to develop a rigorous understanding of the real number, convergence, and properties of single-variable real-valued functions like continuity and differentiability.
- Class materials: Lecture notes, worksheet
- After class: Read the first three pages of Section 1.2. This is material we likely won’t have time to discuss during lecture, but I’m happy to talk about in office hours or on the question and answer forum. Work on Homework 0.
Friday
- Topic: Section 1.2: Preliminaries. We discuss the triangle inequality and get refreshed on some basic proof techniques.
- Class materials: Lecture notes, worksheet
- After class: Read the first two pages of Section 1.3. Work on Homework 1.
Friday
Chapter 1
Monday
- Topic: Section 1.3: The Axiom of Completeness. We introduce the notion of the supremum of a set and the key defining property that distinguishes \(\mathbb R\) from \(\mathbb Q\).
- Class materials: Lecture notes, worksheet
- After class: Read the first three pages of Section 1.4.
Wednesday
- Topic: Section 1.4: Consequences of Completeness. We discuss some properties of the real numbers, like the Archimedean Property and the density of \(\mathbb Q\) in \(\mathbb R\), that come about due to the Axiom of Completeness.
- Class materials: Lecture notes, worksheet
- After class: Read the first two pages of Section 1.5. For tomorrow’s quiz, you should know the definitions of the upper bound of a set, the maximum of a set, and the supremum of a set. You should also be prepared to do an induction proof like the ones in Homework 0.
Friday
- Topic: Section 1.5: Cardinality. We discuss what it means for a set to be countable or uncountable.
- Class materials: Lecture notes
- After class: Read Section 2.1, which gives some good motivation for the things we’ll study in the rest of Chapter 2. Work on Homework 2.
Chapter 2
Monday
- Topic: Section 2.2: The Limit of a Sequence. We begin our discussion on limits of sequences by formally defining what it means for a sequence to converge. To help our understanding we’ll practice proving certain example sequences converge.
- Class materials: Lecture notes, worksheet
- After class: Read the first two pages of Section 2.3.
Wednesday
- Topic: Section 2.3: The Algebraic Limit Theorem. We begin our discussion on general theorems about convergent sequences. The focus today is building new convergent sequences from given ones through algebraic operations. We prove familiar facts like the limit of a sum is the sum of the limits, provided the limits exist.
- Class materials: Lecture notes, worksheet
- After class: Read the end of Section 2.3 on Limits and Order.
Friday
- Topic: Section 2.3: The Order Limit Theorem. We discuss inequalities involving limits of convergent sequences.
- Class materials: Lecture notes, worksheet
- After class: Read Section 2.4. Work on Homework 3.
Chapter 2
Monday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Wednesday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Friday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Chapter 2
Monday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Wednesday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Friday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
TBA
Monday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Wednesday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Friday
- Topic: Exam 1.
- After class:
TBA
Monday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Wednesday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Friday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Spring Break
Monday
- Topic: Spring break, no class.
Wednesday
- Topic: Spring break, no class.
Friday
- Topic: Spring break, no class.
TBA
Monday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Wednesday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Friday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
TBA
Monday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Wednesday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Friday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
TBA
Monday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Wednesday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Friday
- Topic: Exam 2.
- After class:
TBA
Monday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Wednesday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Friday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
TBA
Monday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Wednesday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Friday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
TBA
Monday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Wednesday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Friday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Review
Monday
- Topic:
- Class materials: Lecture notes, worksheet
- After class:
Getting help
Here are a few ways to get help:
- Office hours: tentatively Mondays & Wednesdays 4:00-5:00, Thursdays 1:00-2:00; additional availability by appointment
- TA help: Our class will have a TA who will be holding evening help. Details will be posted on Moodle.
- Study groups: Other students in the class are a wonderful resource. I want our class to feel like a community of people working together.
- Message board: I’ve set up a question and answer forum for asking and answering questions about class material, including homework problems. My plan is to check the forum regularly and answer questions as they come up, but my hope is everyone in the class will pitch in and answer others’ questions as another form of participation in the class.
Resources
- I’ve collected some resources to help you with some basics of LaTeX in case you’re interested in typesetting your work. This is optional.
- Here is a LaTeX template file for writing nicely formatted documents, along with its pdf output.
- A quick reference is available for commonly used symbols.
- Overleaf is a cloud service that lets you edit and compile LaTeX through a web browser so that no local installation is needed. The free version will be sufficient for our class.
- To install LaTeX locally on your personal computer, I recommend installing MacTex if you use a Mac, MikTeX for Windows, or TeXLive for Linux.
- There are many real analysis books out there. Reading our required textbook will be enough to get a strong understanding of the material in class, but if you’re curious, here are some other good books. The first two are available for free as e-books when signed into the campus network or VPN.
- Elementary Analysis: The Theory of Calculus by Kenneth A. Ross. Similar to our textbook, including lots of details in explanations and good problems for extra practice.
- Real Mathematical Analysis by Charles C. Pugh. Nicely written, but with more emphasis on topology and metric spaces than our textbook.
- Principles of Mathematical Analysis by Walter Rudin. A classic textbook that is a bit terse for a first analysis course but worth reading if you really like the material and have a lot of coursework under your belt. It’s filled with great, challenging exercises.