Instructor: Tim Chumley
Office: Clapp 423
Phone: 538-2525
e-mail: tchumley
Office Hours: Mondays & Wednesdays 3:15-4:15; Tuesdays & Thursdays 4:15-5:15; by appointment

Textbook: Elementary Analysis: The Theory of Calculus by Kenneth A. Ross, ISBN: 9781461462705;
available as a free e-text when on the campus network.


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.

  • Group work. Some of your homework will be done in teams.
    • These problems will be due on Fridays.
    • You should type solutions in LaTeX and submit only one copy on behalf of the whole team.
  • Solo work. A selection of problems will be assigned to be written up individually and turned in each week.
    • These problems will also be due Fridays.
    • You may work with others but the writing should be done on your own.
    • Feel free to write solutions by hand or type them in LaTeX.
  • Gradescope. Homework will be turned in through Gradescope.
    • Please add yourself to the Gradescope course using Entry Code 6PE8EX.
    • Gradescope has made a short tutorial on submitting homework.
  • Collaboration. I want you to work together on the written homework, of course on the group assignments but even on the solo assignments. The process of explaining your ideas to one another really helps in learning math. However, you must write up the solo assignments on your own and avoid copying others’ work directly. Please acknowledge sources of help, including classmates or outside textbooks, by writing “help from …” at the top of your assignment. 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!
    • Your revisions will be due on Fridays. Please resubmit (only the problems you’re revising) on Gradescope.
    • For some problems it’s difficult to allow redo’s fairly, so on occasion I might not allow a problem to be redone, but this is rare.
Assignment Due
Homework 0 Sep 3
Homework 1 Sep 10
Homework 2 Sep 17
Homework 3 Sep 24
Homework 4 Oct 1
Homework 5 Oct 8
Homework 6 Oct 22

Quizzes

There will be (mostly) weekly quizzes that will be given on Wednesdays. The purpose of these is to check in to see that you’re comfortable with fundamental material like definitions, theorems, and important examples and proofs. Problems will always be related to the previous week’s homework and class topics.

Quiz Date Material
Quiz 1 Sep 8 Week 1
Quiz 2 Sep 15 Week 2
Quiz 3 Sep 22 Week 3
Quiz 4 Sep 29 Week 4
Quiz 5 Oct 6 Week 5
Quiz 6 Oct 13 Week 6
Quiz 7 Oct 27

Exams

There will be a midterm and a final. The date for the mid-term is subject to change slightly.

Exam Due Format Material
Exam 1 Oct 15 in-class, take-home Sections 1-12
Exam 2 Dec 9-13 TBA TBA

Course plan

Our plan is to cover parts of the textbook chapters 1 – 4. If there’s time, we’ll talk about parts of chapter 5 and 6 too. 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.

Introduction


Monday
  • Topic: Sections 1, 2: The natural, rational, and real numbers. A hint at why we need real numbers and some review of induction.
  • Class materials: Lecture notes
  • After class: Read Theorems 3.2 and 3.4 about inequalities; read Definitions 4.1, 4.2, and 4.3 about min/max, boundedness, and sup/inf; try Exercises 1.9, 3.6.
Wednesday
  • Topic: Section 3: Basic inequalities. We discuss the triangle inequality and get started talking about bounded sets.
  • Class materials: Lecture notes, Basic inequalities
  • After class: Read about the Completeness Axiom and the Archimedean Property in Section 4. Work on the Suprema and infima worksheet.
Friday
  • Topic: Section 4: The completeness axiom. A look at sup, inf, and the key defining property of the reals.
  • Class materials: Lecture notes, Suprema and infima
  • After class: Read Theorem 4.7 (Denseness of Q).

Sequences


Monday
  • Topic: Labor Day, no class.
Wednesday
  • Topic: Section 4: The completeness axiom, continued. We discuss the idea that Q is dense in R and talk a little more about the supremum.
  • Class materials: Lecture notes, Q is dense in R
  • After class: Read Examples 1 and 2 and Definition 7.1 in Section 7, and do Exercise 7.3.
Friday
  • Topic: Sections 7: Introduction to sequences and limits. A look at the formal definition of convergence and how we prove sequences converge.
  • Class materials: Lecture notes, Introduction to sequences
  • After class: Read Examples 3 and 4 in Section 8.

Sequences


Monday
  • Topic: Section 8: Discussion about proofs. We continue our look at introductory proof examples involving limits of sequences.
  • Class materials: Lecture notes, More on sequences
  • After class: Work on Homework 2 in preparation for Wednesday’s class.
Wednesday
  • Topic: Section 8: Discussion about proofs. We continue working on introductory proofs involving sequences.
  • Class materials: Lecture notes, A little more on sequences
  • After class: Read Theorems 9.1, 9.2, and 9.3 in Section 9.
Friday
  • Topic: Section 9: Limit theorems. A look at how to prove the theorems we used in calculus for computing limits.
  • Class materials: Lecture notes, Limit theorems
  • After class: Read up through Example 2 in Section 10.

Sequences


Monday
Wednesday
  • Topic: Section 10: The monotone convergence theorem. We continue our discussion of the monotone convergence theorem and work on group homework.
  • Class materials: Lecture notes
  • After class: Read Theorems 11.3, 11.4, and 11.5.
Friday
  • Topic: Section 11: Subsequences. We begin discussing the notion of subsequences, and discuss the deeply important Bolzano-Weierstrass theorem, which says that every bounded sequence has a convergent subsequence.
  • Class materials: Lecture notes
  • After class: Read the definition of Cauchy sequences in Section 10 and read Lemmas 10.9 and 10.10.

Sequences


Monday
  • Topic: Section 10: Cauchy sequences. We introduce another useful class of sequences that always converge.
  • Class materials: Lecture notes
  • After class: Work on Homework 4 in preparation for Wednesday homework time.
Wednesday
  • Topic: Section 10: Cauchy sequences. We continue our discussion on Cauchy sequences.
  • Class materials: Cauchy sequences
  • After class: Read pages 60-62 and 73-76 on limsup, liminf, and subsequential limits.
Friday
  • Topic: Sections 10, 11: Limit supremum and limit infimum. Not all sequences converge, but it’s possible to define notions of limits even when they oscillate.
  • Class materials: Lecture notes
  • After class: Read Theorems 11.7, 11.8, 11.9.

Sequences and series


Monday
  • Topic: Sections 11, 12: Accumulation points and subsequential limits. We relate the notions of liminf, limsup and subsequential limits.
  • Class materials: Lecture notes
  • After class: Work on Homework 5
Wednesday
Friday
  • Topic: Section 14: Series. We discuss infinite series, looking at and proving some familiar tests of convergence from calculus.
  • Class materials: Lecture notes
  • After class: Enjoy fall break! Start studying for exam.

Series


Monday
  • Topic: Fall break, no class.
Wednesday
  • Topic: Section 14: Series. We continue our discussion of infinite series, proving the comparison theorem and looking at some examples.
  • Class materials: Lecture notes, Series
  • After class: Continue your study for the exam.
Friday
  • Topic: Exam. We’ll spend the entire day on the in-class portion of the exam.
  • After class: Work on take-home portion of the exam.

Series


Monday
  • Topic: Section 14, 15: Series, continued. We finish the worksheet from last time and begin to discuss series involving negative terms.
  • Class materials: Lecture notes
  • After class: Read about the ratio test in Section 14 and alternating series test in Section 15.
Wednesday
  • Topic: Section 14, 15: Series, continued. We discuss the ratio test and alternating series.
  • Class materials: Lecture notes, More on series
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Continuity


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Continuity


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Continuity


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TBA


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TBA


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Wednesday
  • Topic: November break, no class.
Friday
  • Topic: November break, no class.

TBA


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Wrap up


Monday
  • Topic: Wrap up.
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Getting help

Here are a few ways to get help:

  • Office hours: Mondays & Wednesdays 3:15-4:15; Tuesdays & Thursdays 4:15-5:15; by appointment
  • Study groups: Other students in the class are a wonderful resource. I want our class to feel like a community of people working together.
  • Piazza: I’ve set up an online forum for asking and answering questions about class material, including homework problems. My plan is to check the forum every day and answer questions as they come up, but I hope everyone in the class will pitch in and answer others’ questions when possible.

Resources

  • When away from campus, a college VPN is available to access resources normally only available when on the campus network (eg. free e-books through LITS and Springer Link). However, I understand there might be some accessibility issues even with the VPN, so I will do my best to help everyone access necessary materials through other methods as well.
  • I’ve collected some resources to help you when writing homework solutions with LaTeX.
    • Here is a LaTeX template file for writing homework solutions and 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.
    • Understanding Analysis by Stephen Abbott. Similar to our textbook, but with some interesting chapter introductions and projects.
    • 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 too terse for a first analysis course but worth reading if you’re really excited about analysis. It’s filled with great, challenging exercises.