**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.)

Week | Monday | Tuesday | Wednesday | Thursday | Friday |

0 | Sep 29 | Oct 2: No homework due | Oct 3: Introduction | ||

1 | Oct 6: S1, Natural numbers, induction | Oct 8: S2, Integers, rationals | Oct 9: Homework 1 due | Oct 10: S10, Cauchy sequences of rationals | |

2 | Oct 13: S3,S4,S5, Real numbers | Oct 15: Sets and functions | Oct 16: Homework 2 due | Oct 17: Cardinality of sets | |

3 | Oct 20: Countable and uncountable sets | Oct 22: S7,S8 Sequences and convergence | Oct 23: Homework 3 due | Oct 24: S9,S10,S12 Limit points, lim sup, lim inf | |

4 | Oct 27: Midterm #1 | Oct 29: S14, Standard sequences, series, absolute convergence | Oct 30: Homework 4 due | Oct 31: S15, Convergence tests | |

5 | Nov 3: S15, Root and ratio tests | Nov 5: S11,Subsequences, Bolzano-Weierstrass theorem | Nov 6: Homework 5 due | Nov 7: S20, Limiting values of functions | |

6 | Nov 10: S17, Continuity | Nov 12: S18, Maximum principle, intermediate value theorem | Nov 13: Homework 6 due | Nov 14: S19, Uniform continuity | |

7 | Nov 17: S28, Differentiability | Nov 19: S28, Properties of differentiable functions | Nov 20: Homework 7 due | Nov 21: Midterm #2 | |

8 | Nov 24: S32, Riemann integral definition | Nov 26: S33, Riemann integral, existence | Nov 27: No class | Nov 28: No class | |

9 | Dec 1: S33, Riemann integral, properties | Dec 3: S29, Mean value theorem | Dec 4: Homework 8 due | Dec 5: S34, Fundamental theorem of calculus | |

10 | Dec 8: Catch up, review | Dec 10: Catch up, review | Dec 11: Homework 9 due | Dec 12: Catch up, review |

**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.

- (Solutions removed)

- (Solutions removed)