Math 140A, Fall 2022

Foundations of Real Analysis I

## Announcements:

##

## Course Information

Foundations of Real Analysis I

- During finals week I will have office hours Monday 3-5 and
Tuesday 10-11 and Juno will have office hours Tuesday 12:00-4:00 (HSS
4016).

- During the week of the second midterm, my office hours will be Monday 11-12 and 2-4.

- I'm sick today (Nov. 7) and am doing everything remote. Lecture notes for recorded Lecture Nov. 7 (video on Canvas). My office hours are at zoom id 976 2314 0201.
- On Wednesday Nov. 9 my office hours will be 11:00-11:30 and 4:00-4:30 (rather than 11:00-12:00).

- During the week of the first midterm, my office hours will be Monday 11-12 and 2-4.

Professor: Brandon Seward (pronouns: they/them or he/him)

Email: bseward@ucsd.edu

Lecture: MWF 9:00-9:50 in Ridge Walk Academic Building (RWAC) room 0121

Lecture Recordings: On our Canvas page under the Media Gallery tab

Office Hours: Mon. 2-4 and Wed. 11-12 in AP&M 5739

Teaching Assistant: Juno Seong

Email: jseong@ucsd.edu

A01 Discussion: Mon. 5:00-5:50 in Center Hall room 217B

A02 Discussion: Mon. 6:00-6:50 in Center Hall room 217B

Office Hours: Tues. 12:00 - 4:00 in Humanities and Social Sciences Building (HSS) 4016

Course Description: First course in a rigorous three-quarter sequence on real analysis. Topics include: the real number system, basic topology, numerical sequences and series, continuity.

Textbook: Principles of Mathematical Analysis by Walter Rudin, 3rd edition. We will cover Chapters 1 through 4 (excluding the appendix to Chapter 1).

Additional Study Materials: (you will not be tested on these)

Homework: Homework will be due most weeks on Wednesdays at 11:59 PM, except on weeks in which we have a midterm it will be due on Friday at 11:59 PM. No late homework will be accepted, but your lowest homework score will be dropped when computing your final grade. On each assignment, a few problems will be graded for correctness, while the others will be graded simply for completion. We will use Gradescope for turning in homework. When registering for gradescope, please register using your "@ucsd.edu" email address and use Entry Code 4VDVEN.

Homework 1 (Due Wednesday Oct. 5): Chapter 1 problems 1, 4, 5, 6, 7, 9. But in 6(c) change the definition of B(x) to require t < x (instead of t <= x), and solve 6(c) by applying 7(e). This change makes 6(d) easier.

Homework 2 (Due Wednesday Oct. 12): Chapter 1 problems 10, 13, 15, 17; Chapter 2 problems 2, 11; and Problem A: Prove that the set of all injections from the set of natural numbers to itself is uncountable.

Homework 3 (Due Sunday Oct. 23): Chapter 2 problems 6, 7, 8, 9, 10, 14, 22

Homework 4 (Due Friday Oct. 28): Chapter 2 problems 12, 13, 16, 23, 24, 29

Homework 5 (Due Wednesday Nov. 2): Chapter 2 problems 15, 17, 18, 19, 20, 21 and Chapter 3 problem 1 (Hint for problem 18: Mimick the construction of the Cantor set)

Homework 6 (Due Wednesday Nov. 9): Chapter 3 problems 2, 3, 5, 16(a), 17(abc), 20, 23

Homework 7 (Due Sunday Nov. 20): Chapter 3 problems 4, 6, 7, 9, 10, 14(abcd)

Homework 9 (Due Friday Dec. 2): Chapter 4 problems 2, 3, 4, 5, 6, 7

Exams: All exams will be in-person. If you miss a midterm no makeup exam will be given. Instead your final exam will count towards 60% of your final grade.

## Approximate Course Schedule

Email: bseward@ucsd.edu

Lecture: MWF 9:00-9:50 in Ridge Walk Academic Building (RWAC) room 0121

Lecture Recordings: On our Canvas page under the Media Gallery tab

Office Hours: Mon. 2-4 and Wed. 11-12 in AP&M 5739

Teaching Assistant: Juno Seong

Email: jseong@ucsd.edu

A01 Discussion: Mon. 5:00-5:50 in Center Hall room 217B

A02 Discussion: Mon. 6:00-6:50 in Center Hall room 217B

Office Hours: Tues. 12:00 - 4:00 in Humanities and Social Sciences Building (HSS) 4016

Course Description: First course in a rigorous three-quarter sequence on real analysis. Topics include: the real number system, basic topology, numerical sequences and series, continuity.

Textbook: Principles of Mathematical Analysis by Walter Rudin, 3rd edition. We will cover Chapters 1 through 4 (excluding the appendix to Chapter 1).

Additional Study Materials: (you will not be tested on these)

- A list of errata in the course textbook (by George Bergman)

- A supplement to the exercises in the course textbook (by George Bergman)
- Elementary Analysis: The Theory of Calculus by K. A. Ross, 2nd edition (standard textbook for 142A; less advanced but may be helpful. Available for free to UCSD students on the campus network, and can be found by searching in UCSD's Library Catalog).
- Understanding Analysis by Stephen Abbott

- 140A-B Lecture Notes (by Professor Todd Kemp). Note that these lecture notes do not follow our textbook and may cover different topics. This is for reference only.
- Construction of the real numbers by Dedikind cuts - see the appendix to chapter 1 of the course textbook

- Construction of the real numbers by Cauchy sequences (by Professor Todd Kemp - this will make more sense after we discuss Cauchy sequences in class)
- Uniqueness of the real number field and containment of the rationals.

Homework: Homework will be due most weeks on Wednesdays at 11:59 PM, except on weeks in which we have a midterm it will be due on Friday at 11:59 PM. No late homework will be accepted, but your lowest homework score will be dropped when computing your final grade. On each assignment, a few problems will be graded for correctness, while the others will be graded simply for completion. We will use Gradescope for turning in homework. When registering for gradescope, please register using your "@ucsd.edu" email address and use Entry Code 4VDVEN.

Homework 1 (Due Wednesday Oct. 5): Chapter 1 problems 1, 4, 5, 6, 7, 9. But in 6(c) change the definition of B(x) to require t < x (instead of t <= x), and solve 6(c) by applying 7(e). This change makes 6(d) easier.

Homework 2 (Due Wednesday Oct. 12): Chapter 1 problems 10, 13, 15, 17; Chapter 2 problems 2, 11; and Problem A: Prove that the set of all injections from the set of natural numbers to itself is uncountable.

Homework 3 (Due Sunday Oct. 23): Chapter 2 problems 6, 7, 8, 9, 10, 14, 22

Homework 4 (Due Friday Oct. 28): Chapter 2 problems 12, 13, 16, 23, 24, 29

Homework 5 (Due Wednesday Nov. 2): Chapter 2 problems 15, 17, 18, 19, 20, 21 and Chapter 3 problem 1 (Hint for problem 18: Mimick the construction of the Cantor set)

Homework 6 (Due Wednesday Nov. 9): Chapter 3 problems 2, 3, 5, 16(a), 17(abc), 20, 23

Homework 7 (Due Sunday Nov. 20): Chapter 3 problems 4, 6, 7, 9, 10, 14(abcd)

Homework 8 (Due Wednesday Nov. 23): Chapter 3 problems 8, 11(a), 19, 21

Homework 9 (Due Friday Dec. 2): Chapter 4 problems 2, 3, 4, 5, 6, 7

Exams: All exams will be in-person. If you miss a midterm no makeup exam will be given. Instead your final exam will count towards 60% of your final grade.

- First
Midterm. [Practice Midterm]
Wednesday Oct. 19. Will cover Chapter 1 and the first two sections of
Chapter 2 ("Finite, Countable, and Uncountable Sets" and "Metric
Spaces").

- Second Midterm. [Practice Midterm]
Wednesday Nov. 16. Will cover the following sections from the book:
Compact Sets, Perfect Sets, Connected Sets, Convergent Sequences,
Subsequences, Cauchy Sequences, Upper and Lower Limits, and Some
Special Sequences

- Final Exam. [Practice Final] Wednesday Dec. 7 from 8:00 AM to 11:00 AM.
Will cover Chapter 1 (excluding the appendix), Chapter 2, Chapter 3,
and the first two sections of Chapter 4 (Limits of Functions and
Continuous Functions).

- 20% Homework, 20% First Midterm, 20% Second Midterm, 40% Final Exam.
- 20% Homework, 20% Best Midterm, 60% Final Exam

Week |
Monday |
Wednesday |
Friday |

0 |
September 23 Ordered sets |
||

1 |
September 26 Ordered sets Fields |
September 28 Fields |
September 30 The Real field The extended Real number system The Complex field |

2 |
October 3 The Complex field Euclidean spaces |
October 5 (HW 1 Due) The Complex Field, Euclidean spaces |
October 7 Euclidean spaces Finite, countable, and uncountable sets Metric spaces |

3 |
October 10 Metric spaces |
October 12 (HW 2 Due) Metric spaces |
October 14 Metric spaces |

4 |
October 17 Compact sets |
October 19 First Midterm Practice Midterm |
October 21 (HW 3 Due Sunday) Perfect sets |

5 |
October 24 Connected sets Convergent sequences |
October 26 Convergent sequences Subsequences Cauchy sequences |
October 28 (HW 4 Due) Cauchy sequences Upper and lower limits |

6 |
October 31 Upper and lower limits Some special sequences Series |
November 2 (HW 5 Due) Series Series of non-negative terms The number e |
November 4 The number e The Root and Ratio Tests |

7 |
November 7 Power series Summation by parts Absolute convergence |
November 9 (HW 6 Due) Addition and multiplication of series Rearrangements |
November 11 Veterans Day Holiday |

8 |
November 14 Rearrangements Limits of functions |
November 16 Second Midterm Practice Midterm |
November 18 (HW 7 Due Sunday) Limits of functions |

9 |
November 21 Continuous functions |
November 23 (HW 8 Due) Continuous functions Continuity and compactness |
November 25 Thanksgiving Holiday |

10 | November 28 Continuity and compactness Continuity and connectedness |
November 30 Discontinuities Monotonic functions |
December 2 (HW 9 Due) Monotonic functions Infinite limits and limits at infinity |

11 |
Wednesday December 7, 8:00 AM -- 11:00 AM Final Exam Practice Final |