Text: Understanding Analysis (2nd edition) by Stephen Abbott.

Course Description: In M413 we study the real number system and its functions, focusing on topics already familiar from Calculus: 

◦ Convergence of sequences and series

◦ Continuity of functions

◦ Differentiation

◦ Integration

What's new is that we study these topics with complete mathematical rigor. This makes M413 extremely challenging. No undergraduate course at IU is more difficult.  We demand precision and proof in all that we do, and require students to attain a level of mastery that comes only with prolonged effort and practice. While M413 is “elementary” in that it develops calculus from a small number of axioms and definitions, “elementary” does not mean “easy.”  M413 emphasizes skill in reasoning and communication. Though supremely mathematical, it is a language art more than a computational one.

The payoff for the hard work is a remarkable sharpening of the mind’s eye: the ability to see an unexpected and intricate level of detail and structure in the real number system and in scientific models based upon it.

Mastery of precise definitions is a critical requirement for success in this course. Each word in a definition—and their order—tends to be pivotal. Precise memorization of definitions is imperative for success in this course. Students who invest time in memorizing definitions as soon as they're stated will find that their effort pays off richly. Students who don’t will quickly find themselves lost in a frustrating fog of confusion.

If you work hard enough to succeed in M413, you will enjoy a profoundly new and deeper understanding of mathematics. In mastering elementary analysis, one gets a satisfying grasp of the vocabulary and machinery of calculus—a glimpse behind the curtain of physical reality and into its elegant abstract structure. This glimpse cannot be had by any of the physical senses; it is only visible with the kind of pure thinking we develop and practice in M413.

Homework: Homework exercises are the heart of M413, and I will assign some after each lecture. Generally, exercises I assign in a given week will be due at the beginning of class on Wednesday of the next week. After each set is turned in, I will select 7 of the assigned problems for grading on a 2 point scale, so that each week’s homework set is worth 14 points. I will not accept late homework unless I have given advance permission.

Homework must be submitted in hardcopy (no email attachments, please). Write it up in a clear, organized way. The “answers” to M413 exercises usually resemble essays more than calculations. You can't (and should not) get partial credit if the grader can’t follow your reasoning.

I encourage you to collaborate with classmates on homework—assuming of course that you do your share of the thinking—but you must write up your solutions in your own words.  If you don't, you will miss the process of inventing and formulating your own reasoning. If you don't practice these skills on the homework, you certainly won't be able to demonstrate them on the exams.

Please: staple homework pages together before submitting.

We will have 12 or more homework assignments, but when the semester ends, I will keep only your 10 highest scores. The highest possible total homework score is thus 140 points.

Exams: I plan to give one quiz, two in-class midterm exams, and a  2-hour final exam.

The quiz will cover basic definitions to help motivate you to learn them—and help me assess your progress with the material. It will probably be given around the end of week 4 (September 20); I will announce the official date a week in advance.

 

Please mark these exam dates in your calendars:

Midterm 1:	Monday, Sep 30       (Week 6)	In class	120 points
Midterm 2:	Friday, Nov 8          (Week 11)	In class	120 points
Final Exam:	Friday, Dec 20	12:30—2:30pm (room TBA)	200 points

 

Office Hours: My office, Rawles Hall 361, will be open every 

• Tuesday   1:30 – 2:30p

• Thursday  2:45 – 3:45p

Please drop by during these times for help with homework or any other course-related concerns during these times – no appointment needed.  I enjoy working with students one-on-one in office hours, and I hope to meet everyone that way before the semester advances too far. 

 

Grading Assistant: My assistant for this course will grade the homework and will help me grade exams. For questions or concerns about grading, however, please see me.

 Grades: Students accumulate course points from homework and exams. Approximately 610 points will be available, as follows:

 

Item	Unit value	Total value
Homeworks	10 highest scores, 14 pts each	140 points
Quiz	30 points	30 points
Midterm exams	120 points each	240 points
Final exam (2 hours)	200 points	200 points

 

After the final exam, I will rank students' total point accumulations and assign letter grades. I’ll start with the standard curve (lowest A- = 90%, lowest B- = 80%, etc). If that turns out to be too stringent by traditional M413 standards, I’ll relax it. I guarantee the curve will be no tougher than the standard scale.

 

Courtesy and Integrity: Our text contains all the information students need to succeed in the course, so I don’t require class attendance. Experience shows, however, that students who skip the lectures do poorly on exams. I work hard to make lectures worthwhile, and I ask: If you choose to attend a lecture, please arrive on time. Late arrivals distract the class and compromise the learning atmosphere I strive to create.

For the same reasons, please silence and stow all mobile devices during class.

I strive to foster a culture of trust and fairness in my classes. Most students work extremely hard to succeed, and I am glad to go the extra mile to help those students. Any student who attempts to succeed by cheating, however—in any way— completely disrespects those efforts. So I have no tolerance for cheating. The university grants me the authority to impose any penalty for academic dishonesty, up to and including an F course grade. As my record shows, I will not hesitate to exercise that option if I discover cheating. I sincerely hope the need never arises in our class.

 

Check your scores/Monitor my records: I'll post and update the Canvas gradebook after each assignment and each test. You can inspect/check my records and monitor your progress there.

 

Final note: All information here is subject to revision. Check back frequently for updates.

 

M413 Learning Outcomes: 

• The ability to use precise quantifiers (like "for each" and "there exists") to formulate and prove simple statements about sequences of real numbers and their convergence or divergence.
• A working command of cardinality and topological properties of sets of real numbers (e.g., countability and uncountability, open/closedness, compactness).
• A working command of the definition of continuity of a function, along with an understanding of its basic consequences, including the Extreme and Intermediate Value Theorems.
• A working command of the definition of the derivative of a function, along with an understanding of its basic consequences, including the Mean Value Theorem
• The ability to define the Riemann Integral, and to detect/prove integrability (or its lack) for basic functions.
• A working command of the Fundamental Theorem of Calculus.