Math 4111
Introduction to Analysis, Fall 2022
Course Description
The real number system and the least upper bound property; metric spaces (completeness, compactness, and connectedness); continuous functions (in ℝn; on compact spaces; on connected spaces); C(X) (pointwise and uniform convergence; Weierstrass approximation theorem); differentiation (mean value theorem; Taylor's theorem); the contraction mapping theorem; the inverse and implicit function theorems. Prerequisite: Math 310 or permission of instructor.
Contact Information
Instructor: Ari Stern
Email: stern@wustl.edu
Office: Cupples I 211B
Office Hours: Tu 11am-12:30pm, Th 2-3:30p
Lectures
Section 1: MWF 10-10:50am, Seigle 103
Section 2: MWF 3-3:50pm, Cupples II 200
The first class will be on Monday, August 29, and the last will be on Friday, December 9. Class will be canceled for Labor Day (Monday, September 5), Fall Break (Monday, October 10) and Thanksgiving Break (Wednesday, November 23 through Friday, November 25).
Homework Assignments
Problem sets will be posted to Canvas approximately weekly, and students will submit their solutions electronically by uploading them before the specified due date and time. You are encouraged to discuss the homework with your fellow students, but your final write-up must be your own. Please make sure that your solutions are written clearly and legibly.
Anthony Hong (hong.x@wustl.edu) and Mingzhen Li (l.mingzhen@wustl.edu) are responsible for grading the homework assignments.
Exams
There will be one in-class midterm exam on Wednesday, October 19. The final exam will be held Thursday, December 15, from 3:30pm-5:30pm.
Note: The final exam was originally scheduled at different days/times for Section 1 (December 19, 10:30am-12:30pm) and Section 2 (December 15, 6-8pm), but the registrar changed this to a common exam day/time for both sections. If you have an exam conflict at the new time, please contact the instructor ASAP.
Grading
Grades will be based on a weighted average of homework (40%, lowest score dropped), midterm exam (20%), and final exam (40%).
Required and Supplemental Texts
The required textbook for this course is Principles of Mathematical Analysis, by Walter Rudin (third edition, McGraw Hill, 1976). Feel free to use either a physical or electronic copy of this text, as long as it is this edition.
This book, widely known as "baby Rudin" has been the standard undergraduate real analysis text for 70 years and has trained generations of mathematicians. It also has a well-deserved reputation for being terse, leaving the reader occasionally wishing for a bit more explanation, motivation, intuition, etc. I'll try to provide some of this in lectures, of course—but it can also be helpful to see different presentations of the same material, and this is generally the case when learning mathematics, regardless of the text or topic.
Here are a few optional texts to supplement Rudin. All are available for free download from the campus network; off campus, you'll need to use the WUSTL Library Proxy.
- Real Mathematical Analysis, by Charles C. Pugh
- Elementary Analysis, by Kenneth A. Ross
- Understanding Analysis, by Stephen Abbott
Pugh covers similar material to Rudin, and the book has a breezy, conversational tone. Ross and Abbott mainly stick to analysis on the real line—they don't go into as much depth or generality as Rudin and Pugh regarding analysis on other spaces—but they give outstanding treatments of the material they do cover.
I have also asked the library to place the following text on reserve:
- The Real Analysis Lifesaver, by Raffi Grinberg
This book was written as a study companion for the first few chapters of Rudin. It also reviews some of the foundational material you may remember from Math 310 or 203 (logic, set theory, methods of proof, etc.).
Academic Integrity
All students are expected to adhere to high standards of academic integrity, as specified in the undergraduate student academic integrity policy. Since this course is offered through the College of Arts & Sciences, any violations of this policy will be referred to the College’s Academic Integrity Officer. Violations of this policy include, but are not limited to:
- homework collaboration exceeding that permitted in the Homework Assignments section above (e.g., preparing your final write-up side-by-side with another student rather than independently, copying another student's solutions);
- submitting the work of others as your own (e.g., copying solutions found online);
- exam misconduct (e.g., copying another student's exam or knowingly allowing your exam to be copied, giving or receiving unauthorized assistance, use of unauthorized books/notes or other materials, unauthorized use of electronic devices, etc.).
If you have any questions, or are unsure about what is permitted/prohibited by this policy, please ask me.
In many cases, academic integrity violations are the result of getting behind in coursework and making bad decisions under pressure. Keep up with your assignments, ask questions when you are unsure what is expected of you, and do not give in to the temptation to cut corners.
COVID-19 Health and Safety Protocols
Students experiencing symptoms consistent with COVID-19 or concerned about a possible exposure should contact Habif Health and Wellness Center (314-935-6666) to arrange for testing as indicated. If instructed by Habif to quarantine or isolate, students should notify their instructor as soon as possible by forwarding the email they received from Habif. Any accommodation needs for COVID-related absence not covered in an instructor’s standard course policies should be discussed between the student and instructor.
While on campus, it is imperative that students follow all public health guidelines established to reduce the risk of COVID-19 transmission within our community. The full set of University protocols can be found at https://covid19.wustl.edu/health-safety/.
Access to Classroom Content During Excused Absences (Including COVID)
To encourage in-person attendance and participation, WashU no longer requires that lectures be recorded or otherwise made available outside the classroom, aside from certain exceptional cases: "In the case of excused student absences due to COVID and other factors (e.g., illness, religious holidays, family emergencies, etc.), instructors should develop strategies for providing students access to the fundamental content of a given class session so that students are able to make progress in the course while complying with public health and university guidelines around quarantine and isolation and managing other challenges that disrupt their ability to attend class."
If you cannot attend class due to COVID or other factors similar to those listed above, and if you contact me at least 2 hours prior to the start of class, I will try to arrange for video recording of the lecture. If I am unable to record the lecture (e.g., due to technical difficulties) or do not receive advance notice to do so, I will provide written lecture notes instead.