Analysis 1, Spring 2025

Course: Math 425A, Analysis, Spring 2025
Prerequisite: MATH 290 or MATH 430 or MATH 432
Course Content: The real number system, metric spaces, limits, continuity, derivatives and integrals, infinite series.
Last update: 19 December 2024

Instructor: Steven Heilman, stevenmheilman(@-symbol)gmail.com
Office Hours: Fridays, 1015AM-12PM, KAP 406G
Lecture Meeting Time/Location: Mondays, Wednesdays, and Fridays, 1PM-150PM, KAP 147

TA: Jishnu Bose, jishnubo(@-symbol)usc.edu
TA Office Hours: held in the Math Center
Discussion Section Meeting Time/Location:

Tuesdays, 10AM-1150AM, KAP 147

Textbook: The following textbook is recommended but not required.
Tao, Analysis I, Hindustan Book Agency, 2006. 2nd Ed.

Other textbooks: R.S. Strichartz, The Way of Analysis, 2000. Revised Ed.

Exam 1: Friday, February 21, 1PM-150PM, KAP 147
Exam 2: Wednesday, March 26, 1PM-150PM, KAP 147
Final Exam: Wednesday, May 7, 2PM-4PM, KAP 147
Other Resources: An introduction to mathematical arguments, Michael Hutchings, An Introduction to Proofs, How to Write Mathematical Arguments

Email Policy:

My email address for this course is stevenmheilman@gmail.com
It is your responsibility to make sure you are receiving emails from stevenmheilman@gmail.com , and they are not being sent to your spam folder.
Do NOT email me with questions that can be answered from this syllabus.

Exam Procedures: Students must bring their USCID cards to the midterms and to the final exam. Phones must be turned off. Cheating on an exam results in a score of zero on that exam. Exams can be regraded at most 15 days after the date of the exam. This policy extends to homeworks as well. All students are expected to be familiar with the USC Student Conduct Code. (See also here.)
Accessibility Services: If you are registered with accessibility services, I would be happy to discuss this at the beginning of the course. Any student requesting accommodations based on a disability is required to register with Accessibility Services (OSAS) each semester. A letter of verification for approved accommodations can be obtained from OSAS. Please be sure the letter is delivered to me as early in the semester as possible. OSAS is located in 301 STU and is open 8:30am-5:00pm, Monday through Friday.
https://osas.usc.edu/
213-740-0776 (phone)
213-740-6948 (TDD only)
213-740-8216 (fax)
OSASFrontDesk@usc.edu

Discrimination, sexual assault, and harassment are not tolerated by the university. You are encouraged to report any incidents to the Office of Equity and Diversity http://equity.usc.edu/ or to the Department of Public Safety http://capsnet.usc.edu/department/department-public-safety/online-forms/contact-us. This is important for the safety whole USC community. Another member of the university community - such as a friend, classmate, advisor, or faculty member - can help initiate the report, or can initiate the report on behalf of another person. The Center for Women and Men http://www.usc.edu/student-affairs/cwm/ provides 24/7 confidential support, and the sexual assault resource center webpage sarc@usc.edu describes reporting options and other resources.

Exam Resources: Here and here are pages with past exams for a related course. Here are some exams from when I taught related courses: Exam 1 Exam 1 Solution Exam 2 Exam 2 Solution Final Final Solution Exam 1 Exam 1 Solution Exam 2 Exam 2 Solution Final Final Solution

Homework Policy:

Homeworks are due at 10AM Tuesdays.
Homeworks are submitted in brightspace. To avoid internet issues, I recommend making your first submission of an assignment well in advance of the deadline.
Homeworks should be submitted as single PDF documents. One way to create a PDF document from paper homework assignments is the freely available Adode Scan App
Late homework is not accepted.
If you still want to turn in your homework late, then the number of minutes late divided by ten will be deducted from the homework score. The exact deduction and rounding procedure is not guaranteed to be accurate.
The two lowest homework scores will be dropped. This policy is meant to account for illnesses, emergencies, dropped internet connections, etc.
You may not use the internet to try to find answers to homework problems.
Do not submit homework via email.
Collaboration on homeworks is allowed and encouraged.
All homework assignments must be written by you, i.e. you cannot copy someone else`s solution verbatim.

Grading Policy:

The final course grade is weighted as the larger of the following two schemes.
- Scheme 1: class participation (3%), homework (17%), the first midterm (20%), the second midterm (25%), and the final (35%).
- Scheme 2: class participation (3%), homework (17%), the largest midterm grade (30%), and the final (50%).
The grade for the semester will be curved. However, I do not "curve down" since anyone who exceeds my expectations in the class by showing A-level performance on the exams and homeworks will receive an A for the class.
If you cannot attend one of the exams, you must notify me within the first two weeks of the start of the quarter. Later requests for rescheduling will most likely be denied.
Class participation is not the same as attendance. I will never explicitly take attendance, but I will notice if someone is frequently absent. Things that increase your class participation grade include: asking good questions, paying attention in class, showing up on time or early to class, etc. Things that decrease your class participation grade include: excessive talking or disruptions during class, frequent absences, excessive texting/smartphone usage in class, frequent tardiness, etc.
You must take the final exam to pass the course.

Tentative Schedule: (This schedule may change slightly during the course.)

Week	Monday	Tuesday	Wednesday	Friday
1	Jan 13: Introduction	Jan 14: No homework due	Jan 15: S2, Integers, rationals	Jan 17: S10, Cauchy sequences of rationals
2	Jan 20: No class	Jan 21: Homework 1 due	Jan 22: S3, S4, S5, Real numbers	Jan 24: Sets and functions
3	Jan 27: Cardinality of sets	Jan 28: Homework 2 due	Jan 29: Countable and uncountable sets	Jan 31: S7, S8 Sequences and convergence
4	Feb 3: S9, S10, S12 Limit points, lim sup, lim inf	Feb 4: Homework 3 due	Feb 5: S14, Standard sequences, series, absolute convergence	Feb 7: S15, Convergence tests
5	Feb 10: S15, Root and ratio tests	Feb 11: Homework 4 due	Feb 12: S11, Subsequences, Bolzano-Weierstrass theorem	Feb 14: S20, Limiting values of functions
6	Feb 17: No class	Feb 18: Homework 5 due	Feb 19: S17, Continuity	Feb 21: Midterm #1
7	Feb 24: S18, Maximum principle, intermediate value theorem	Feb 25: No homework due	Feb 26: S19, Uniform continuity	Feb 28: S28, Differentiability
8	Mar 3: S28, Properties of differentiable functions	Mar 4: Homework 6 due	Mar 5: S32, Riemann integral definition	Mar 7: S33, Riemann integral, existence
9	Mar 10: S34, Fundamental theorem of calculus	Mar 11: Homework 7 due	Mar 12: S33, Riemann integral, properties	Mar 14: S29, Mean value theorem
10	Mar 17: No class	Mar 18: No homework due	Mar 19: No class	Mar 21: No class
11	Mar 24: Integration by parts	Mar 25: No homework due	Mar 26: Midterm #2	Mar 28: Integration by parts
12	Mar 31: Change of variables	Apr 1: Homework 8 due	Apr 2: Metric Spaces	Apr 4: Metric Spaces
13	Apr 7: Cauchy sequences	Apr 8: Homework 9 due	Apr 9: Compactness	Apr 11: Continuity
14	Apr 14: Continuity	Apr 15: Homework 10 due	Apr 16: Sequences of Functions	Apr 18: Uniform convergence
15	Apr 21: Uniform convergence	Apr 22: Homework 11 due	Apr 23: Series of Functions	Apr 25: Uniform approximation by polynomials
16	Apr 28: Power Series	Apr 29: Homework 12 due	Apr 30: Exponential and Logarithm	May 2: Review of Course

Advice on succeeding in a math class:

Review the relevant course material before you come to lecture. Consider reviewing course material a week or two before the semester starts.
When reading mathematics, use a pencil and paper to sketch the calculations that are performed by the author.
Come to class with questions, so you can get more out of the lecture. Also, finish your homework at least two days before it is due, to alleviate deadline stress.
Write a rough draft and a separate final draft for your homework. This procedure will help you catch mistakes.
If you are having difficulty with the material or a particular homework problem, review Polya's Problem Solving Strategies, and come to office hours.

Homework

HW1 HW2 HW3 HW4 HW5 HW6 HW7 HW8 HW9 HW10 HW11 HW12 soln

Homework .tex files

HW1 HW2 HW3 HW4 HW5 HW6 HW7 HW8 HW9 HW10 HW11 HW12

Exam Solutions

Exam 1 Exam 1 Solution Exam 2 Exam 2 Solution Final Final Solution

Supplementary Notes

425a Lecture Notes