Math 131 A – Real Analysis, Winter 2020

Midterm: Friday Feb 7th, in class.

Final: Thursday, March 19th,8:00am-11:00am, Location TBD

Lectures: MWF 1pm-1:50 pm in MS 5138.

Office Hours: Wed 4:00pm–5:00pm and Friday 10–11am in 5626 Math Sciences Building. Classroom office hour Thursday 4-5pm MS 6118 (Times subject to change, NO THURSDAY OFFICE HOUR DURING FIRST WEEK.)

Overview of the course: This is a course about basic Real Analysis, introducing you to a rigorous
treatment of analysis with a full appreciation of the pathologies and difficulties one may run into. Our goal is to discuss
and define concepts, such as real numbers, continuity, limits, derivatives and integrals. We will learn to argue about them precicely
and appreciate the importance of assumptions, theorems and preciceness.

You might ask why is this a required course for a math major, and why one might want to take it even if it is not required. Or perhaps,
why would you take it, if you never intend to go to mathematics graduate programs. One general reason is that, while you may have
learned of proofs, in this class we will focus on rather complex statements. These are a degree more complex as they often involve
quantifiers, such as for all, and there exists, and implications between them. Understanding these well is useful in theoretical
computer science, and wherever you need to justify your conclusions. Thus, we will place a lot of emphasis on reading, understanding
and writing proofs. The context will be that of calculus. Even here, in this rather simple context, we will learn to appreciate that assumptions matter.

Another reason is to gain an appreciation for the value of approximation and estimation. Numerical analysis, optimization and
data analysis rest on the idea of approaching a correct solution. Good algorithms come with error bounds and justifications for them=.
To understand them well, and to appreciate them, one needs some basic abilities to argue and manipulate these. Thus, you should hopefully
also walk away with an ability to estimate and bound quantities, and to understand when things are “sufficiently small”.

Course Book: K.A. Ross, Elementary Analysis: The Theory of Calculus, 2nd Ed. The sections below will refer to this book. Other good references: 1) Abbott, Understanding Analysis, 2) Rudin, Principles of Real Analysis, 3) (Oldie) Whittaker, Watson: A course of modern analysis.

Skills to learn and that will be tested:

• Enhance ability to read proofs, statements and understand them.
• Understand how to verify conditions of a statement.
• Define limits, continuity, derivatives and integrals.
• Prove basic features of real numbers using axioms. Derive conclusions from a restricted set of assumptions.
• Be able to prove, for simple examples, existence of limits, or continuity, integrability or differentiability.
• Prove properties of derivatives, e.g. how they behave under sums, products, compositions.
• Understand the least upper bound property, and featurers of sup, inf, limsup and liminf. Manipulate, give examples.
• Write proofs and learn good proof writing skills: clarity, preciceness and explaining your reasoning.

Instructor: Sylvester Eriksson-Bique, 5262 Math Sciences Building.

TAs:

• Alan Zhou: 4A Tuesday 1pm-1:50pm

Exams: One in-class midterm: Friday Feb 7th, in class. Three-hour final: Thursday, March 19th 8am–11am.

Bring student ID to both midterms and the final. There will be no make-up exams.
No calculators, notes, or books will be permitted in any exam.

Homework: There will be bi-weekly homework. It is due in the beginning of the class on Friday. Further information is given below.

• No late homework will be accepted.
• Graded for a total of 20 points each set, 5 for each graded problem (usually three) and 5 for completeness of remainder.
• Write your name, ID number.
• Staple your pages! May submit online as pdf on CCLE, but please remember to SUBMIT.
• The weakest homework score will be omitted.

Discussion sections: Discussion sections will include a roughly weekly quiz in the beginning. This will be followed by a discussion on the proof, as well as discussing each others solutions. Lowest quiz
score will be dropped. Attendance required for the quiz.

Grading:

• Normal: Discussion quizzes: 10%; Homework, 20%; Midterm 30%; Final 40%.
• Optional (no quiz): Discussion quizzes: 0%; Homework, 20%; Midterm 40%; Final 40%.
• Given a compelling (and documented) reason to miss the midterm: Quiz 10%; Homework, 20%; Final 70%.
• Final exam can not be missed. If missed, it will lead to an incomplete or F, and require retaking class/final in a different quarter.

Extra credit: At any point in the class, you can present a detailed proof to me during office hours. Schedule this ahead of time.
Depending on interest, during the last three weeks of class there will be additional times available to present. I will give
a list of problems you can choose from and you should present a complete proof on the board for one of them. Credit is 1-3 % depending on
the quality of the presentation. You can retake once to increase credit. 1% is acceptable, and a valid proof with minor flaws
that we can clarify, 2 % complete proof but somewhat disorganized or flawed, 3 % is perfect except at most minor flaws. Half credit
can also be given for attempt.

Inclusivity: This course is welcoming to everyone and from any background. If you at any time feel
a policy of this course causes excessive hardship or is difficult to satisfy for whatever reason (e.g. office hours
conflict with work, or exams conflict with religious holiday), please do let me know ahead of time and we will
work to find an accommodation. If you encounter problems in your personal life, or medical issues, please do reach out for help,
and I can also do my best to forward you to university resources that can help you.

CAE brochure

My goal is to foster an inclusive, supportive and encouraging enviroment for you to learn. Many of us suffer from various degrees
of learning difficulties, exam stress etc. Please be aware that the university has many resources to help with challenges you might face
and that I as an instructor will do my best to assist and accomodate different learners. Please do not be afraid to reach out to a councelor,
me or your TA for contact information to various services. Also, please review the attached brochure.

https://www.cae.ucla.edu/learning-disabilities-brochure