**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 will be posted on the Course Announcements Moodle forum throughout the semester, but essentially all other materials will be posted on this page.

Check the syllabus for all the important class policies (grades, attendance, etc.).

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.

- These problems will be due on
**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.

- These problems will also be due
**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.

- Please add yourself to the Gradescope course using Entry Code
**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.

- Your revisions will be due on

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 |

Homework 7 | Oct 29 |

Homework 8 | Nov 5 |

Homework 9 | Nov 12 |

Homework 10 | Nov 19 |

Homework 11 | Dec 3 |

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 29 | Week 7 |

Quiz 8 | Nov 3 | Week 8 |

Quiz 9 | Nov 12 | Week 9 |

Quiz 10 | Dec 1 | Week 12, 13 |

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 |

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.

**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.

**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.

**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).

**Topic**: Labor Day, no class.

**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.

**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.

**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.

**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.

**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.

**Topic**: Section 10: The monotone convergence theorem. We show that sequences that are bounded and either increasing or decreasing must always converge.**Class materials**: Lecture notes, Monotone convergence theorem**After class**: Work on Homework 3.

**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.

**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.

**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.

**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.

**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.

**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

**Topic**: Sections 11, 12: Accumulation points and subsequential limits. We wrap up our discussion.**Class materials**: Lecture notes, Subsequential limits and limit supremum and infimum**After class**: Read about Cauchy criterion, test for divergence, and comparison test in Section 14.

**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.

**Topic**: Fall break, no class.

**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.

**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.

**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, Series worksheet solutions**After class**: Read about the ratio test in Section 14 and alternating series test in Section 15.

**Topic**: Section 14, 15: Series, continued. We discuss the ratio test and alternating series.**Class materials**: Lecture notes, More on series, (worksheet solutions)**After class**: For Monday, read Definition 17.1, Theorem 17.2, and Examples 1 and 2 in Section 17.

**Topic**: Leap symposium, no class.

**Topic**: Sections 17, 20: Limits of functions. We begin our discussion of limits of functions and continuity and get introduced to epsilon-delta proofs.**Class materials**: Lecture notes**After class**: Read Examples 1 and 3 in Section 17. Work on the group homework problem as practice and preparation for next time.

**Topic**: Section 17, 20: Limits and continuity. We discuss what it means for a function to be continuous at a point, and do more examples of epsilon-delta proofs.**Class materials**: Lecture notes**After class**: Read Example 3 and Theorems 17.3 and 17.4.

**Topic**: Section 17, 20: Limits and continuity. We discuss more proofs involving limits of functions with the aim of distinguishing which definition is appropriate to use and when.**Class materials**: Lecture notes**After class**: Read Theorem 18.1 and its proof.

**Topic**: Section 18: Properties of continuous functions. We discuss the extreme value theorem.**Class materials**: Lecture notes**After class**: Read Theorem 18.2 and its proof.

**Topic**: Section 18: Properties of continuous functions. We discuss the intermediate value theorem.**Class materials**: Lecture notes**After class**: Read through to Definition 19.1 in Section 19.

**Topic**: Section 19: Uniform continuity. We discuss a stronger form of continuity where our choice of \(delta\) is independent of \(x_0\)**Class materials**: Lecture notes, Uniform continuity, worksheet solutions**After class**: Read Examples 3 and 4 and Theorem 19.2 in Section 19.

**Topic**: Section 19: Uniform continuity. We discuss why continuous functions on closed intervals must be uniformly continuous, as well as how to prove a function fails to be uniformly continuous.**Class materials**: Lecture notes, More on uniform continuity, worksheet solutions**After class**: Read Definition 28.1 and try Exercises 28.1ac, 28.3.

**Topic**: Section 28: Differentiability. We introduce the idea of differentiable functions and how to prove a function is differentiable at a point.**Class materials**: Lecture notes**After class**: Read Theorems 28.2 and 28.3.

**Topic**: Section 28: Differentiability. We discuss some theorems that are a consequence of differentiability.**Class materials**: Lecture notes**After class**: Read Theorems 29.1, 29.2, and 29.3.

**Topic**: Section 29: Mean Value Theorem. We build up a proof for one of the most important results in the theory of calculus.**Class materials**: Lecture notes, Three core theorems, worksheet solutions**After class**: Read Corollaries 29.4 and 29.5 and Theorems 34.1 and 34.3.

**Topic**: Section 34: Fundamental Theorem of Calculus. We use uniform continuity and the mean value theorem in the context of integration.**Class materials**: Lecture notes, Fundamental Theorem of Calculus**After class**: Read Definitions 24.1, 24.2, and Example 2 in Section 24.

**Topic**: Section 24: Pointwise convergence. We talk about sequences of functions and introduce a simple, but in a sense flawed, notion of convergence for these sequences.**Class materials**: Lecture notes, Pointwise convergence, worksheet solutions**After class**: Read Theorem 24.3, Remark 24.4, and Example 7 in Section 24.

**Topic**: Section 24: Uniform convergence. We discuss a new notion of convergence and a computational technique for verifying it.**Class materials**: Lecture notes, Uniform convergence, worksheet solutions**After class**: Enjoy the break!

**Topic**: November break, no class.

**Topic**: November break, no class.

**Topic**: Section 24, 25: Uniform convergence. We discuss general properties of uniformly convergent sequences of continuous functions.**Class materials**: Lecture notes, Uniform convergence and integrals**After class**: Work on Homework 11 and read Theorem 25.2. Study for the quiz on definitions of pointwise and uniform convergence and proofs of corollaries to the mean value theorem.

**Topic**: Section 25: Uniform convergence. We discuss the worksheet from last time and answer the question of interchanging limits and integrals.**Class materials**: Uniform convergence and integrals, worksheet solutions**After class**: Work on Homework 11.

**Topic**: Section 25: Uniform convergence. We discuss interchanging limits and derivatives.**Class materials**: Uniform convergence and derivatives, worksheet solutions**After class**: Do some review of material from the second half of the semester.

**Topic**: Wrap up.**Class materials**:**After class**: Study! Enjoy the break! Keep in touch!

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.

- 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.