## Math 112: Introduction to Analysis, Spring 2015

Section S01: MTWF 8-8:50am, Library 389
Office Hours: TW 11-noon, θ 2:30-3:30pm, Library 313

Math
Center: SuMTWθ 7-9pm, Library 387

Textbook: *Introduction to Analysis* by Irena Swanson.

Compiled course notes: notes.pdf. Click on a day of the week to get notes for that day.

### Week 1: Jan. 26 - 30

- Monday: Review the syllabus. Logical statements.
- Tuesday: Read §1.1. Manipulations of statements. Proofs.
- Wednesday: Read §1.2. Quantifiers. More proofs.
- Friday: Read §1.3. More proof methods, negation.

#### Homework due Friday

- 1.1.4, 1.1.5, 1.1.10, 1.1.12*, 1.2.1, 1.2.3(i,ii) [SOLUTIONS]

Any problems marked with a * are optional

### Week 2: Feb. 2 - 6

- Monday: Read §1.4. Summation and mathematical induction.
- Tuesday: Read §§1.5-1.6. Mathematical induction and Pascal's triangle.
- Wednesday: Read §2.1. Quiz on §§1.1-1.5 [SOLUTIONS]. Sets.
- Friday: More about sets.

#### Homework due Friday

- 1.2.6, 1.3.11(v), 1.3.12, 1.3.15, 1.3.16, 1.4.1, 1.5.2, 1.5.3, 1.5.7, 1.5.10, 1.5.20, 1.5.28, 1.5.31(ii)*, 1.5.33* [SOLUTIONS]

### Week 3: Feb. 9 - 13

- Monday: Read §§2.2-2.3. Cartesian products, relations, and equivalence relations.
- Tuesday: Reread §2.3. More on relations and equivalence relations.
- Wednesday: Read §2.4. Quiz on §§1.6-2.3 [SOLUTIONS]. Functions.
- Friday: Read §2.5. Binary operations on sets.

#### Homework due Friday

- 1.6.2, 1.6.9*, 2.1.1, 2.1.3(vii), 2.1.8, 2.2.1 (for the second part, use 2.1.1), 2.3.1, 2.3.2, 2.3.4, 2.3.7, 2.3.10*, 2.4.1, 2.4.4, 2.4.6 [SOLUTIONS]

### Week 4: Feb. 16-20

- Monday: Reread §2.5. More on binary operations.
- Tuesday: Read §3.1. Inductive sets. [VIDEO]
- Wednesday: Read §3.2. Quiz on §§2.4-2.5, 3.1 [SOLUTIONS]. Arithmetic.
- Friday: Read §3.3. Order. Constructing the integers.

#### Homework due Friday

- 2.4.5, 2.4.7, 2.5.2, 2.5.3, 2.5.5, 2.5.6, 2.5.7, 2.5.8*, 3.1.2, 3.1.3, 3.1.4*, 3.2.1, 3.2.3 [SOLUTIONS]

### Week 5: Feb. 23-27

- Monday: Read pp.103-108. Arithmetic and order on the integers.
- Tuesday: Read pp.109-113. The rational numbers.
- Wednesday: Read pp.77-85. Quiz on §§3.2-3.6 [SOLUTIONS]. Fields and ordered fields.
- Friday: Read pp.114-123. The real
numbers. (Notes include the conclusion of the proof of Claim 2:
*U*= 2.)^{2}

#### Homework due Friday

- 3.5.1, 3.5.5, 3.5.6, 3.6.2, 3.6.6, 3.6.7, 2.6.3, 2.6.5*, 2.6.6, 2.6.7, 2.6.8, 2.6.9, 2.7.13, 2.7.16 [SOLUTIONS]

### Week 6: March 2-6

- Monday: Read pp.124-127. The complex numbers.
- Tuesday: Read pp.128-133. The complex norm.
- Wednesday: Modular arithmetic redux.
- Friday: Review.

#### Homework due Friday

- 3.8.3, 3.8.4, 3.9.1, 3.9.2, 3.9.3, 3.9.4, 3.9.7 [the statement of part (iii) is incorrect in the printed notes; figure out what the correct statement is and prove that one!], 3.9.10, 3.10.4, 3.10.5(iii), 3.10.8 [do this one by hand or with a computer]. [SOLUTIONS]
- Review problems. You don't need to turn these problems in, but next week's exam will include at least two of these problems.

### Week 7: March 9-13 (guest instructor: Angélica Osorno!)

- Monday: 60-minute take-home exam: exam1.pdf
- Tuesday: Read pp.134-139. Topology and density.
- Wednesday: Read pp.143-155. Limits: to be or not to be.
- Friday: Read pp.156-161. Limit theorems.

#### Homework due Friday

- 3.11.1, 3.11.5, 4.1.1, 4.1.5, 4.1.8, 4.2.2 [SOLUTIONS]

### Week 8: March 16-20

- Monday: Read §5.1. Continuous functions.
- Tuesday: Read §5.2. The intermediate and extreme value theorems.
- Wednesday: Skim Chapter 6. Quiz on (material covered from) Chapters 4 and 5 [SOLUTIONS]. Review of differentiation.
- Friday: Read §6.5. Higher order differentiation and Taylor polynomials.

#### Homework due Friday

- 4.3.5, 4.3.9, 4.3.11, 5.1.2, 5.1.4, 5.1.5, 5.1.6, 5.1.7, 5.2.3, 5.2.6, 5.2.7 [SOLUTIONS]

### Spring Break: March 21-29

- Review document on the algebra and topology of the complex
numbers: topologyC.pdf. These problems
are not required, but they are
*highly*recommended, and we will continue to use this content throughout the rest of the semester.

### Week 9: March 30 - April 3

- Monday: Read §8.1. Sequences.
- Tuesday: Read §8.2. Convergence.
- Wednesday: Read §8.3. Quiz on continuity and basic sequences. [SOLUTIONS] Divergence and infinite limits.
- Friday: Read §8.4. Convergence via functions.

#### Homework due Friday

- 8.1.1, 8.1.4, 8.1.5, 8.1.7, 8.2.1, 8.2.4, 8.2.7, 8.2.10, 8.2.12*, 8.3.1, 8.3.4, 8.3.6, 8.3.8(v) [(partial) SOLUTIONS]

### Week 10: April 6-10

*Announcement*: Brunch for gender minorities in math and physics,
April 11. RSVP to rhokaner@reed.edu or nuxolla@reed.edu by Monday,
April 6.

- Monday: Read §8.6. Bounded sequences, monotone sequences, the ratio test for sequences.
- Tuesday: Read §8.7. Cauchy sequences.
**R**and**C**are Cauchy complete. - Wednesday: Quiz on §§8.4, 8.6, 8.7. Experiments with distance.
- Friday: Read §8.8. Subsequences.

#### Homework due Friday

- 8.5.13(i,ii,vi), 8.5.14, 8.5.20(ii,iii), 8.5.21* (the problems
in §8.5 can be solved by appealing to limit theorems from
§8.4; it's unlikely that you'll need an ε-
*N*proof), 8.6.1(ii), 8.6.1(vi)*, 8.6.2, 8.6.7, 8.6.12, 8.7.2, 8.7.4 [SOLUTIONS]

### Week 11: April 13-17

- Monday: Read §9.1. Infinite series.
- Tuesday: Read §9.2. Convergence tests.
- Wednesday: Quiz on §§8.8, 9.1, 9.2 [SOLUTIONS]. Complete metric spaces, complete fields.
- Friday: Read §9.3. Power series and radius of convergence.

#### Homework due Friday

- 8.8.2*, 8.8.4, 9.1.1, 9.1.2, 9.1.4, 9.1.10, 9.2.1(ii,iii,iv), 9.2.5, 9.2.11, 9.2.12 [SOLUTIONS]

### Week 12: April 20-24

- Monday: Read §9.4. Taylor series.
- Tuesday: Read §9.5. Differentiation of power series.
- Wednesday: Review session. (Donuts!)
- Friday: In-class exam [SOLUTIONS]. Empahsis on content covered between Tuesday of Week 7 and Wednesday of Week 11.

#### Review exercises

### Week 13: April 27 - May 1

- Monday: Read §9.7. The complex exponential.
- Tuesday: Read §9.8. Trigonometry and Euler's formula.
- Wednesday: Visualizing the complex exponential. Course evals.
- Friday: Parlor tricks.

#### Homework due Friday

- 9.4.2, 9.4.4, 9.5.1, 9.5.2(i,ii,iii,iv), 9.7.1, 9.7.2, 9.7.6, 9.8.2, 9.8.4 [SOLUTIONS].

### Review session: Thursday, May 7, 1:30-3:30pm, Library 389.

### Final exam: Tuesday, May 12, 1-4pm in Library 389.

### The LaTeX document preparation system

Poor handwriting? Love escape characters? Too much free time? Try LaTeX!

- LaTeX at Reed.
- A short guide [pdf] to writing mathematics with LaTeX.
- Sample LaTeX input/output: sample.tex/sample.pdf.

Kyle M. Ormsby