# Textbook

Real Analysis: A First Course, 2nd ed. by Russell Gordon.

# Course Goals

By the end of the semester, we hope

• to advance your ability to prove abstract mathematical statements.

• to read proofs critically for their explanatory value, truth value, and conciseness.

• to improve your presentation (written and oral) of mathematical proofs.

• to begin to grasp the richness of the real numbers.

• to develop a foundational understanding of sequences, limits, functions, continuity, differentiation, and integration over the real numbers.

• to devise proof strategies on your own and to write proofs of mathematical statements on your own (concerning the topics above) that you have not seen before.

# Classroom Community

Mathematics is a highly collaborative and social enterprise. We learn better when we learn together. In order to achieve our goals, we must foster mutual respect, regardless of background or beliefs. Racism, sexism, or other forms of discrimination have no place in the classroom or at the College. All students are capable of success, and it is imperative that we work under that ethos. Practice empathy and humility.

# Homework (30%)

There will be regular homework assignments. Your overall homework average will comprise 30% of your grade.

The week's homework exercises will be posted on the syllabus. While changes may be made to the week's assignment as the week progresses, ample notice will be given.

The problems on the Detailed Syllabus and are to be written up according to the style guidelines. All of the problems are to help you learn the material. Writing up the problems helps to solidify your understanding and provides you a written record of exercises to aid your exam studies.

Above & Beyond. Each write-up problem must have a modest extension that you formulate and include on your own, in addition to the assigned work. There must also be a brief written description of the piece of the assignment that you consider to be the Above & Beyond component (so that I'll know what to look for). This requirement will be worth roughly 10% of any particular exercise grade.

Upload & $\LaTeX$ Your written work should be typeset using $\LaTeX$. The free, on-line site sharelatex allows you to typeset documents using $\LaTeX$ .

Once you have written your homework, convert it to a pdf document and upload it to me.

Late Policy. If your homework is $d$ days late, then your grade will be docked by $\frac{d(d+1)}{2}$ percentage points.

# Participation (20%)

Presenting mathematical ideas to a group is an important part of doing mathematics. Mathematicians don't cloister themselves in their rooms and do mathematics in complete isolation. More commonly, mathematicians talk through questions, ideas, proofs and conjectures with other mathematicians. It can be social and fun.

To encourage this, I expect each student to present something in class at least once each week. I also expect each student, when not presenting, to engage the presenter by asking questions, making comments, and offering criticism. Each week I'll post a grade that assesses your participation for the week. It'll be vague, subjective and mysterious. The average participation grade will count for 20% of your overall grade.

This excerpt from the the Instructor's Resource Manual that accompanies the book "Closer and Closer: Introducing Real Analysis" by Carol Schumacher encapsulates the kind of participation I'll be looking for in the class.

# Pre-class (10%)

Individual class periods are designed with the assumption that you have read the day's material in advance. A typical class will have us going over some discussion problems, reviewing some of the more esoteric ideas in the readings, and presenting proofs. To support this activity most days will have a short pre-class reading or exercise. You can find the link to the pre-class topics on the class syllabus next to the reading for the day. Remember it's timestamped and is due at least one hour before the start of class.

No late pre-class work will be accepted.

Your average score on pre-class work is worth 10% of the final grade.

# Quizzes (15%)

There will be several short in-class quizzes. Typically you'll be presented with a single statement to prove. You will be able to use your book and notes during the quizzes.

# Exams (25%)

There will be several exams during the semester (roughly one per chapter) and one final exam. The final exam will be cumulative.

The exam scores will be averaged and equally weighted (except for the final exam). The final exam will receive double the weight of an ordinary exam.

Letter grades will be assigned according to the following scale:
A - 92-100
A- - 90-91
B+ - 88-89
B - 82-87
B- - 80-81
C+ - 78-79
C - 72-77
C- - 70-71
D+ - 68-69
D - 62-67
D- - 60-61
F - 0-59

# Cooperation vs. Cheating

• Cooperation... not collaboration. Real Analysis can be a difficult topic. The only way to get it is to do it yourself. As such, while you may discuss homework problems with each other and with me. You should not discuss them with Google.

• Once you have a solution, or idea, you must write it up on your own. All the work you submit to me is implicitly understood to be your own unless otherwise specified.