If you would like to review the fundamentals of reading and writing proofs, here are some recommendations:

• Develop the ability to read mathematics with a critical and inquisitive mind, and use these skills to both refine and strengthen your learning process.
• Develop technical writing skills, including the ability to communicate detailed and precise mathematical concepts correctly, clearly, and efficiently.
• Learn how to understand, use, and manipulate complicated mathematical definitions that involve several nested quantifiers.
• Appreciate the history of how analysis developed, together with the motivation behind the definitions we adopt, by understanding some counterintuitive examples of sequences and functions.
• Learn some important concepts of analysis, including supremums, sequences, limits, series, topology of the real line (open, closed, and compact sets), continuity, pointwise convergence, uniform convergence, and Riemann integration.

Homework assignments will be due on Fridays, and will be posted to the course webpage.

• Homework is due at the beginning of class.
• Your lowest homework score will be dropped.
• Solutions to the homework will be posted to the course webpage.
• Please take the time to write your solutions neatly and carefully! All homework assignments consisting of more than one page must be stapled.
• Unless you have a serious emergency that you bring to my attention before a homework assignment is due, late homework will not be accepted for credit. However, please feel free to take advantage of the fact that one homework score is dropped to skip writing up a homework assignment if you have more pressing demands on your time.

Although there will certainly be some "computational" problems in the course, most of the homework involves writing proofs and/or detailed explanations. This means that there are often many correct answers. It also means that the clarity of exposition and the proper use of mathematical terminology and notation are as vital to your solutions as having the correct idea. A major goal of this course is to develop your mathematical writing skills to point where you can write detailed, clear, and even sometimes elegant proofs. Do not be alarmed if your homework has many comments! An important part of my job is to give you suggestions and guidance for how to improve.

I strongly recommend that you learn to type your solutions. LaTeX is a wonderful free typesetting system which produces high-quality documents at the cost of only a small amount of additional work. If you plan to do any kind of scientific writing in the future, you will most likely use LaTeX, so taking the time right now to familiarize yourself with it will pay off.

There will be two in class exams and a scheduled three hour final exam, each of which will focus on conceptual problems and proofs.

In class exams dates: October 8 and November 19.
Final exam date: Thursday, December 20 at 2:00pm.

On Mondays without an exam, there will be a short quiz focusing on definitions and examples. These quizzes are low stakes (each counts for about 1% of the grade), and are designed to encourage you to regularly review the foundational material of the course. In addition, they will provide diagnostic assessments (for both of us!) to help determine where to focus our time. Your lowest quiz score will be dropped.

In order to learn mathematics effectively, it is essential to constantly ask questions, to isolate what aspects of the material are unclear, and to make conjectures. To help develop these skills, I strongly encourage you to participate in class and also to bring questions to office hours. Furthermore, before the start of each Wednesday class, you should send me (by email) at least two questions that you have about the subject matter of the course. For example, you can ask whether a conjecture you have is true, how to overcome an obstacle in a proof, or why a definition takes the form it does. Please make the question as specific as you can! Saying "I don't understand the proof of Theorem 2.4.3" is not sufficient. Explain why you are having difficulty with the proof, isolate where you are getting stuck, and ask a question that might help clarify your understanding.

Consult the general Grinnell College policy on Academic Honesty and the associated booklet for general information.

Homework: If you enjoy working in groups, I strongly encourage you to work with others in the class to solve the homework problems. If you do collaborative work or receive help form somebody in the course, you must acknowledge this on the corresponding problem(s). Writing "I worked with Sam on this problem" or "Mary helped me with this problem" suffices. You may ask students outside the course for help, but you need to make sure they understand the academic honesty policies for the course and you need to cite their assistance as well. Failing to acknowledge such collaboration or assistance is a violation of academic honesty.

If you work with others, your homework must be written up independently in your own words. You can not write a communal solution and all copy it down. You can not read one person's solution and alter it slightly in notation/exposition. Discussing ideas and/or writing parts of computations together on whiteboards or scratch paper is perfectly fine, but you need to take those ideas and write the problem up on your own. Under no circumstances can you look at another student's completed written work.

You may look at other sources, but you must cite other books or online sources if they provide you with an idea that helps you solve a problem. However, you may not specifically look for solutions to homework problems, and you may not solicit help for homework problems from online forums.

Quizzes, Exams, and Final: You may neither give nor receive help. Books, written notes, computers, phones, and calculators are not permitted at any time during a testing period.

• Spend time actively reading the book, the course notes, your notes, and any other sources you choose. I read one chapter of my Abstract Algebra book nine times during the course of the semester (not counting reading portions of it to help with homework problems). This is normal, and you should not expect to master a proof or a concept by simply going to class and reading about it once.

• When reading the book or the course notes, do not sit down and read linearly like you would a novel. Have a pencil and paper at hand, and work through the arguments and examples on your own. After reading a proof, stop to process it in your own way and then work through it again in your own words. When you see an example illustrating a result, go back through the proof in the context of that one example.

• There will be many new concepts, definitions, and examples introduced throughout the course. In my experience, attempting to memorize them directly is not the most useful way to succeed. Instead, the way to master these ideas is to work with them (both on the homework and on your own) until they become comfortable and natural.

• The material is challenging at both a conceptual level and a detail-oriented technical level. In my opinion, the best way to understand both is to regularly switch back and forth between the two. When trying to understand a difficult theorem, read it several times at different levels (first try to understand the overall structure, then break it down into key points, then work through specific details, and finally repeat the whole process). When working through an example, picture it in the context high level theory we are developing.

I encourage students with documented learning, physical, or psychiatric disabilities to discuss appropriate accommodations with me. You will also need to have a conversation with, and provide documentation of your disability to, the Coordinator of Disability Resources, John Hirschman, located on the third floor of the JRC (x3089).

I encourage students who plan to observe holy days that coincide with class meetings or assignment due dates to consult with me as soon as possible so that we may reach a mutual understanding of how you can meet the terms of your religious observance and also the requirements for this course.