Math 3001 (Fall 2024):
Real Analysis 1
About this course
Welcome to Real Analysis 1! This class is a rigorous treatment of calculus, from first principles, with a focus on mathematical proof. Beginning with the natural numbers and sets, we will build up the theory of the real numbers, including sequences, series, limits, convergence, continuity, derivatives, and the Riemann integral.
Contact information
- Instructor: Jonathan Wise
- Office: Math 204
- E-mail: jonathan.wise@colorado.edu
You may also contact me anonymously.
Office hours
My office is Room 204 in the Math Department. My office hours sometimes change, so I maintain a calendar showing the times I will be available. Some of my office hours may be held over Zoom, so please check the calendar before trying to find me in person. You will be able to find the Zoom link on Canvas or Discord. I am often in my office outside of those hours, and I'll be happy to answer questions if you drop by outside of office hours, provided I am not busy with something else. I am also happy to make an appointment if my office hours are not convenient for you.
Syllabus
Here is a proposed weekly schedule of the course. It is all but certain we will not stick to this schedule exactly.
- Week 1 (26 Aug — 30 Aug) : Natural numbers (Ch. 2)
- Week 2 (2 Sep — 6 Sep) : Set Theory (Ch. 3, 8)
- Week 3 (9 Sep — 13 Sep) : Integers and rationals (Ch. 4)
- Week 4 (16 Sep — 20 Sep) : Real numbers (Ch. 5)
- Weeks 5 – 6 (23 Sep — 4 Oct) : Sequences (Ch. 6)
- 2 Oct : Exam 1
- Weeks 7 – 8 (7 Oct — 18 Oct) : Series (Ch. 7, 8)
- 23 Oct : Exam 2
- Weeks 9 – 10 (21 Oct — 1 Nov) : Continuity (Ch. 9)
- Weeks 11 – 12 (4 Nov — 15 Nov) : Differentiation (Ch. 10)
- 20 Nov : Exam 3
- Weeks 13 – 15 (18 Nov — 11 Dec) : Riemann Integral (Ch. 11)
If we wind up moving quickly, we will also cover metric spaces.
Textbook
The main textbook for this course is
Terrence Tao. Analysis I, 4e.
If we get to metric spaces, we will also use
Terrence Tao. Analysis II, 4e.
Both of these texts are available for free through the university's subscription to Springer Link. I suggest downloading the texts while on the university network to ensure you have access when you are not on the network.
Prerequisites
From a logical standpoint, this course has no prerequisites : we develop real analysis from first principles. However, you will need to be comfortable in the language and style of mathematical proofs. Having taken calculus will help with intuition.
Course goals
We will learn quite a few definitions, theorems, and proofs in this class. The following list is meant to give an idea of the different levels of mastery you can achieve with them. The letters should also give a rough idea of how those levels of mastery will correspond to a final grade.
- Know the statement of the definition or theorem.
- Recognize situations in which the definition or theorem applies.
- Be able to follow the proof of the theorem. Understand why the hypotheses in the theorem are necessary. Apply the theorem in novel situations.
- Understand the proof of the theorem well enough to prove similar statements with similar methods.
Homework
Your homework will consist of reading the textbook and completing ungraded problems. None of these will be graded directly, but they are essential preparation for the next class's activities.
Homework assignments can be found on Discord. See Canvas for an invitation.
Academic honesty
I encourage you to consult outside sources, use the internet, and collaborate with your peers. However, there are important rules to ensure that you use these opportunities in an academically honest way.
- Anything with your name on it must be your work, and only your work, and it should accurately reflect your understanding.
- Expect plagiarism to be treated severely. Deliberate plagiarism will be reported to the Honor Code Office.
In this class, almost all graded work will be completed in class, so plagiarism should be a limited concern. However, the above principles still apply, as do the following suggestions :
To avoid plagiarism, you should always cite all resources you consult, whether they are textbooks, tutors, websites, classmates, large language models, or any other form of assistance. Using others' words verbatim, without attribution, is absolutely forbidden, but so is using others' words with small modifications. The ideal way to use a source is to study it, understand it, put it away, use your own words to express your newfound understanding, and then cite the source as an inspiration for your work.
The particular format for your citations is not important, provided that they indicate what the source was and they provide sufficient information for a reader to find the source. Textual references should therefore be specific (include a page number, for example). Internet references should have a working URL.
Questions
Do you have a question or comment about the course? The answer might be in the course policies, on this page. If your question isn't answered in the course policies, please send me an email. Or, if you prefer, you may send me a comment anonymously.
Grading
| Component | Contribution | Dates |
|---|---|---|
| Daily work | 25% | Daily |
| Exam 1 | 15% | 2 Oct |
| Exam 2 | 15% | 23 Oct |
| Exam 3 | 15% | 20 Nov |
| Final exam | 30% | 17 Dec |
The scoring on the assessments will be based on the course goals. Notably, your score will not necessarily be a simple sum of point values from each problem, but will instead be my overall assessment of the degree to which you have achieved the course's goals on the relevant topics.
Missed exams will be handled on a case-by-case basis.
Special accommodations, CoViD-19, classroom behavior, and the honor code
The Office of Academic Affairs officially recommends a number of statements for course syllabi, all of which are fully supported in this class.
If you need special acommodation of any kind in this class, or are uncomfortable in the class for any reason, please contact me and I will do my best to remedy the situation. You may contact me in person or send me a comment anonymously.