Course: MATH 131A, Analysis, Lecture 5, Fall 2014
Prerequisite: MATH 32B, Multivariable Calculus, MATH 33B, Differential Equations.
Recommended course: MATH 115A, Linear algebra.
Course Content: Rigorous treatment of the foundations of real analysis, including construction of the rationals and reals; metric space topology, including compactness and its consequences; numerical sequences and series; continuity, including connections with compactness; rigorous treatment of the main theorems of differential calculus. This course should develop your ability to write rigorous proofs.
Last update: 15 December 2014

Lecture Meeting Time/Location: Monday, Wednesday and Friday, 2PM-250PM, Geology 4645
Instructor: Steven Heilman, heilman(@-symbol)ucla.edu
Office Hours: Mondays, 930AM-1130AM, Fridays, 1030AM-1130AM, MS 7370
TA: Sangchul Lee, sos440(@-symbol)math.ucla.edu
TA Office Hours: Tuesdays, 1PM-230PM, MS 2963
Discussion Session Meeting Time/Location: Thursday, 2PM-250PM, Geology 4645
Required Textbook: Elementary Analysis: The Theory of Calculus, 2nd Ed., K.A. Ross. Note: You can download the textbook from the UCLA library, by searching for the book and looking up the eBook copy, or by searching for Springerlink, and then searching for the book within Springerlink.
Other non-required textbooks: T. Tao, Analysis I, Hindustan Book Agency, 2006. 2nd Ed. R.S. Strichartz, The Way of Analysis, 2000. Revised Ed.
TA Course Website: here
First Midterm: October 27, 2PM-250PM, WGYOUNG 4216
Second Midterm: November 21, 2PM-250PM, BOELTER 5249
Final Exam: December 19, 1130AM-230PM, GEOLOGY 3656
Other Resources: 131AH, Tao, Winter 2003: I would highly recommend reading these lecture notes. These notes are also available in book form, which is cited above. Note that these resources correspond to the honors version of the course, so we will not be covering the material in as much detail. My own lecture notes below are meant to be a more condensed presentation of similar material. So, if you prefer a more thorough treatment, I recommend these notes (and the book by Ross).
An introduction to mathematical arguments, Michael Hutchings, An Introduction to Proofs, How to Write Mathematical Arguments
Exam Procedures: Students must bring their UCLA ID 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.
Exam Resources: Here is a page with past exams for the course. Here is another page with past exams for the course. Note that the content of these other courses may be slightly different.
Here is a list of practice final questions (skip Q1,8,9,14,16,17b,21,24,25). Here are solutions.

Homework Policy:

• Late homework is not accepted.
• The lowest homework grade will be dropped. This policy is meant to account for illnesses, emergencies, etc.
• Do not submit homework via email.
• There will be 9 homework assignments, assigned weekly on Thursday and turned in at the beginning of each discussion session on the following Thursday.
• A random subset of the homework problems will be graded each week. However, it is strongly recommended that you try to complete the entire homework assignment.
• You may use whatever resources you want to do the homework, including computers, textbooks, friends, the TA, etc. However, I would discourage any over-reliance on search technology such as Google, since its overuse could degrade your learning experience. By the end of the quarter, you should be able to do the entire homework on your own, without any external help.
• All homework assignments must be written by you, i.e. you cannot copy someone else's solution verbatim. I would encourage you to understand carefully how the homework solutions work, and how you would find such a solution on your own. Overusing collaborations or search technology should result in poor performance on the exams.
• You are free to use any results in my lecture notes, up to the things that we have covered in class. If you use results from the book that are not in my notes, you must also produce the proof from the book, written in your own words.
• Label your homework with the lecture number and the discussion section number.
• Homework solutions will be posted ...

• The final grade is given by the larger of the following two schemes. Scheme 1: homework (15%), the first midterm (20%), the second midterm (25%), and the final (40%). Scheme 2: homework (15%), the largest midterm grade (35%), final (50%). The final grade will be curved. However, 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.
• We will use the MyUCLA gradebook.
• 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.
• You must attend the final exam to pass the course.

Tentative Schedule: (Sections of the book listed below only approximate what we will cover.) (This schedule may change slightly during the 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. Also, consider typesetting your homework. Here is a template .tex file if you want to get started typesetting.
• If you are having difficulty with the material or a particular homework problem, review Polya's Problem Solving Strategies, and come to office hours.

• HW1 HW2 HW3 HW4 HW5 HW6 HW7 HW8 HW9

• (Solutions removed)

• (Solutions removed)

• 1: Introduction, Natural Numbers, Real Numbers, Least Upper Bound
• 2: Cardinality, sequences, series, subsequences
• 3: Functions, continuity, derivatives
• 4: Riemann sums, Riemann integrals, fundamental theorem of calculus
• Real Analysis, 131A, Fall 2014 (full set of notes)