Instructor: Tim Chumley
Office: Clapp 423
Phone: 413-538-2525
e-mail: tchumley

Course: Math 301, Real analysis

Prerequisites: Math 102 (Calculus II), Math 211 (Linear Algebra), and either Math 206 (Intro to Proofs) or Math 232 (Discrete Math).

Textbook: Understanding Analysis by Stephen Abbott.

About the course: Our course is an introduction to analysis through the lens of trying to understand the underpinnings of calculus. This is a nice way to frame the subject the first time you see it, but in a way, the course and subject is much more about the proof techniques that we’ll learn and the nitty gritty details of the synonymous notions of limits, approximation, and convergence. It’s through this wider lens that it becomes clear why analysis is at the core of many mathematical subjects (eg. probability, mathematical statistics, differential equations, scientific computing), as well as outside disciplines like engineering and economics. It’s really cool, and I hope you end up loving it as much as I do!

Learning goals: During the semester our goal will be to understand the theory of the real numbers and functions at a deep, rigorous level. Central to this will be developing our ability to communicate, particularly through writing proofs. In terms of topics specific to our course, the aim is to cover:

  • field axioms, total orderings, suprema and infima of sets, completeness of the real numbers
  • limits, sequences, and series of real numbers, Cauchy sequences, the Bolzano-Weierstrass theorem
  • continuity, compactness, connectedness, the extreme value theorem, the intermediate value theorem
  • uniform convergence, interchanging the order of limits, and power series
  • (time permitting) the mean value theorem, Taylor’s theorem, the fundamental theorem of calculus.

Attendance: When healthy, I expect everyone to come to each class ready to do math. This means that you should bring paper, pens, pencils, and other equipment that you may need. Before each class please prepare by doing any assigned reading and suggested problems. Please expect to talk in small groups as well as in class discussions. Activities may involve worksheets and informal presentations that will be designed to contribute significantly to our learning.

Participation: Besides attending class and doing homework, I would like everyone to participate actively and frequently since it makes such a difference in your learning. I suggest asking and answering questions during lecture and on our online discussion board, working in groups in and out of class, coming to office hours, responding to feedback surveys, and keeping in touch with me and your classmates.

Homework: There will be weekly homework assignments due, except on weeks when there is an exam.

Quiz: There will be weekly quizzes, except on weeks when there is an exam, with an emphasis on definitions, understanding theorem statements, and doing problems similar to worksheets and homework.

Exams: There will be two midterm exams and a final exam.

Technology: Here are some general remarks on the use of calculators, software, and phones:

  • For all homework, quizzes, and exams, you may use a scientific calculator, but it is not necessary or required (much of our work will involve reasoning and writing rather than computation).
  • Software like Wolfram Alpha or Desmos can be used on homework or other outside the class work without citation.
  • ChatGPT or other large language models (LLMs) should be avoided when doing assignments since one of the fundamental focuses of our class is in learning to reason and explain thought processes through writing. To date, such software also tends to be unreliable when it comes to mathematical reasoning. Especially as you’re learning a new subject, it can be hard to tell if an LLM is leading you astray. Having said all this, I’m curious about the future of such systems; please tell me if you find a cool application related to our class.
  • It’s ok to take photos of the board for note taking, but please don’t post these online.
  • Please keep your phones on Do Not Disturb mode in class, especially during quizzes and exams so as not to disturb others.
  • If you’re unsure whether something is ok to use, please feel free to ask.

Late work, makeups: In general, I ask you to turn homework in by the deadlines and take exams on time because it helps you keep up with the class and it helps me to stay organized. However, I nearly universally say yes to short extensions on initial (ie. non-redo) submissions of homework if you ask. Please just get in touch as soon as possible and suggest how long you think you’ll need (one day, for example, is reasonable). The only caveat is that you’ll get less time to submit redos on late work since redo deadlines are firm. If you miss a quiz or exam, please get in touch as soon as possible. We’ll need to have a longer conversation and make detailed plans if you go through an extended absence or missed deadlines or assessments start to pile up, but my goal is to help anyone who falls behind.

Getting help: Here are some of the resources that will be available:

  • TA help: Our class will have a TA who will be available for help on weeknight evenings. More information on their availability will be announced.
  • Office hours: These times (which will be posted near the top of the class web page) are open drop in sessions where there may be multiple students getting help. It’s expected that you come with specific questions about notes, homework problems, or the text that you have thought about already. If you need to schedule a time to meet with me one-on-one, either to discuss something private or because you can’t make my office hours, please send me an email with times that you are available to meet, and I will find a time that works for both of us.
  • Study groups: Other students in the class are a wonderful resource. I want our class to feel like a community of people working together, so I encourage forming study groups and collaboration, but work to be turned in should be written up individually and on your own.

Grades: Grades will be assigned according to the following weighting:

  • Participation: 5%
  • Homework: 25%
  • Quizzes: 10%
  • Exam 1: 20%
  • Exam 2: 20%
  • Exam 3: 20%

Overall letter grades will be based on a scale no stricter than the usual:

  • 93-100: A
  • 90-93: A-
  • 88-90 B+
  • 83-88: B
  • 80-83: B-
  • 78-80: C+
  • 73-78: C
  • 70-73: C-
  • 67-70: D+
  • 63-67: D
  • 60-63: D-
  • 0-60: F

Academic integrity: It is very important for you to follow the Honor Code in all of your work for this course. Collaboration on homework assignments is encouraged. However, it is important that you only write what you understand, and that it is in your own words. My first instinct is always to trust you, but I cannot give credit for plagiarized work and might have to refer such issues to the deans. If you have any questions about whether something is an Honor Code violation, please ask me.

Students with disabilities: If you have a disability and would like to request accommodations, please get in touch with me. We’ll work together, along with Disability Services, to make sure the class is accessible and equitable.