MATH 112 Introduction to Analysis

Spring 2020
Irena Swanson
L386, extension 7399 (503 517 7399)
Office hours in L315: M, W 2:15-3:30, Th 1:30-3pm, or stop by my office at other times, or make an appointment.

Class: Section 1: MWF 8-8:50 in Library 389.
Section 3: MWF 13:10-14:00 in Physics 240A.

Course textbook: Homework is assigned from the version printed in January: Introduction to Analysis (Spring 2020 version).
There are copies of the Spring 2020 version on 2-hour reserve in the library.

Use this constantly updated and improved version of the textbook for a possibly more streamlined exposition, but do NOT use this version for the homework, as numbering may have changed from the official course notes. Please report to me any further mathematical or typographical errors, any suggestions, or any TeXed solutions.

Possible supplemental material:
• Steven R. Lay: Analysis, With an Introduction to Proof. (I have used this book in the past, with complex number supplements.)
• Ray Mayer: Course Notes for Math 112 (Introduction to Analysis).
• Irena Swanson: Introduction to Analysis, with construction of the number systems. (These are almost the same notes as the official course notes, with differences in chapters 2 and 3. The official course notes take it as fact that the set of real numbers forms an ordered field with the Least upper bound property, making more room for the study of sequences and series. These alternative notes instead construct and prove these assumptions from basic set theory.)
Math center tutoring: Sunday through Thursday (yes, five nights a week), 7-9pm, Library 204. Click here for more information.

Week 1: January 27 - 31
• Monday: Course overview.
Section 1.5: Mathematical induction.
Section 1.4: Summation.
• Wednesday: Bring solved Exercise 1.5.7 to class, in a form to be turned in.
Read Section 1.5 before class.
Section 1.5: Mathematical induction.
Section 1.6: Pascal's triangle.
• Friday: Start reading Section 1.1; complete the reading by Monday.
Section 2.1: Sets.
Homework due (always at the beginning of class): 1.5.2, 1.5.3, 1.5.12, 1.5.13 (OR 1.5.8 instead to not replicate the work from class), 1.5.16.

Week 2: February 3 - 7
• Monday: Be ready for class work on Sections 1.1, 1.5.
By Friday finish reading Section 1.2.
• Wednesday: Section 2.1: Sets.
Section 2.2: Cartesian product.
Homework due: 1.1.3, 1.1.6, 1.1.7 iv), 1.1.8. 2.1.1. Do 1.1.5 for yourself (do not turn it in).
• Friday: Section 2.3: Relations, equivalence relations.
Section 2.4: Functions.
Homework due: 2.1.2, 2.1.3 vii), viii), 2.1.4. Only 8am section: 2.2.1, 2.2.2. Think through 2.1.6, 2.1.7, 2.1.8, possibly with Venn diagrams. Think through 2.2.3.
Quiz over induction (Section 1.5).

Week 3: February 10 - 14
• Monday: Be ready with Section 1.2.
Section 2.4: Functions.
For next week be reading the negation part of Section 1.3.
• Wednesday:
Section 2.5: Binary operations on sets.
Turn in 2.3.1, 2.3.3, 2.3.7 i). Only 1pm section: 2.2.1, 2.2.2. Read and understand 2.2.5, 2.2.6.
• Friday:
Section 2.6: Fields.
Homework due: 2.4.2, 2.4.4, 2.4.5, 2.4.9, 2.4.12, 2.4.13 i), ii).
Quiz over sections 1.1, 2.1.

Week 4: February 17 - 21
• Monday: Section 2.7: Order on sets, ordered fields.
Logical negations and how to write proofs, Section 1.3. Read Section 1.3 ahead of class.
• Wednesday: Section 2.9: Increasing and decreasing functions.
Homework due: 2.5.1 i), ii), 2.5.5 for Z/5Z, 2.5.6, 2.6.6, 2.6.8. (Careful: -x is defined as the additive inverse of x.) Read 2.6.2, 2.6.3.
• Friday: Section 2.10: The Least upper bound property of R.
Homework due: 2.7.1, 2.7.4, 2.7.14, 2.7.15, 2.7.16. 2.8.1. Read 2.8.2.
Quiz on Sections 2.3, 2.4.

Week 5: February 24 - 28
• Monday: Section 2.11: Absolute values.
Review of Chapters 1 and 2 through Section 2.7.
Extra and extended office hours on Monday: 10-11:30am, 2:15-3:45.
• Tuesday:
Extra and extended office hours on Tuesday: 10am-noon, 1-2:45pm.
• Wednesday: Section 3.1: Complex numbers.
Section 3.2: Functions related to complex numbers.
No homework due this week.
No office hours on Wednesday or Thursday.
• Friday: In-class exam over Chapters 1 and 2 through Section 2.7.
No quiz.

Week 6: March 2 - 6
• Monday: Go over the exam.
Section 3.3: Absolute value in C.
Section 3.4: Polar coordinates.
• Wednesday: Section 3.5: Topology on R and C.
Homework due: 2.11.5. 3.1.1, 3.1.2, 3.1.3, 3.2.1.
• Friday: From Sections 3.3 and 3.4: Limit point of a set.
Homework due: 3.3.1, 3.3.2, 3.3.4 iii), 3.4.1, 3.4.4.
Quiz over Sections 2.7, 2.9, and 2.10.

In general it is a good idea to read the material before class: in this way you can follow the lecture part more closely and you can ask for clarifications. If you read the material the night before the homework is due, you are unlikely to have your questions answered in time. (You may want to read my further exhortations on how to study.)

If you wish to typeset your homework with LaTeX, check out LaTeX at Reed.

If you wish to typeset your homework with (plain) TeX, click here for a sample plain TeX template for typesetting. Your browser may want to open the file and run tex on it, which means that you would then see the pdf output file and not the .tex file that created it --- make sure that you download this file as a text file. (On a Mac, after clicking here, check your Downloads folder for "samplehomework.tex".) The output of processing that tex file is this pdf file, which you may want to see for comparison. For your thesis you will probably want to use LaTeX, but plain TeX is cleaner, less verbose, and most TeX commands also work in LaTeX. (The ones that don't work have to do with matrices, and LaTeX complains about "obsolete \over" but it executes it anyway.)