MATH 431 Homework

MATH 431/531 - Basic Analysis I
Fall 2012

Homework & Reading Assignments

Assignments are due at the beginning of class on the day listed.

August

Monday

Tuesday

Wednesday

Friday

27	First Class Day

28	Read the syllabus carefully. Read §1.1: The Irrationality of √2 Exercise: 1.2.1

29	Read §1.2: Some Preliminaries Exercises: 1.2.3, 1.2.6

31	Reread §1.2: Some Preliminaries Exercises: 1.2.7, 1.2.9 531: Using only the axioms for an ordered field (pp. 245-246) prove that

a) if x ≤ y then –y ≤ –x b) –(–1) = 1 c) 0·x = 0 for all x d) –0 = 0 e) (–1)(–1) = 1 f) 0 < 1
g) if 0 < x < y, then 0 < y^–1 < x^–1 h) if x < y and z < 0, then yz < xz i) x² ≥ 0 for all x

September

Monday

Wednesday

Friday

Labor Day

5	Read §1.3: The Axiom of Completeness Exercises: 1.3.2, 1.3.3, 1.3.4, 1.3.6

7	Reread §1.3: The Axiom of Completeness Exercises: 1.3.7, 1.3.8, 1.3.9

10	Read §1.4: Consequences of Completeness Exercises: 1.4.1, 1.4.4 531: Prove that the Nested Interval Property implies the Least Upper Bound Property.

12	Reread §1.4: Consequences of Completeness Exercises: 1.4.5, 1.4.6

14	Reread §1.1-§1.4 Resubmit 3 rewritten exercises from Chapter 1

17	Chapter 1 Exam

19	Read §2.1: Rearrangements of Infinite Series

21	Read §2.2: The Limit of a Sequence Exercises: 2.2.1, 2.2.2, 2.2.4, 2.2.6, 2.2.7

24	Read §2.3: The Algebraic and Order Limit Theorems Exercises: 2.3.1, 2.3.2, 2.3.3

26	Reread §2.3: The Algebraic and Order Limit Theorems Exercises: 2.3.4, 2.3.7, 2.3.8, 2.3.9

27	Read §2.4: The Monotone Convergence Theorem and Infinite Series Exercises: 2.4.2, 2.4.5, 2.4.6

October (tentative)

Monday

Wednesday

Friday

1	Read §2.5: Subsequences and the Bolzano-Weierstrass Theorem Exercises: 2.5.1, 2.5.3 531: Prove that the Monotone Convergence Theorem implies the Least Upper Bound Property.

3	Reread §2.5: Subsequences and the Bolzano-Weierstrass Theorem Exercises: 2.5.4, 2.5.6

5	Read §2.6: The Cauchy Criterion Exercises: 2.6.1, 2.6.2, 2.6.3

8	Reread §2.6: The Cauchy Criterion Exercises: 2.6.5, 2.6.6 531: Prove that the Bolzano-Weierstrass Theorem implies the Least Upper Bound Property.

10	Reread §2.1-§2.6 Study for the Chapter 2 Exam

12	Chapter 2 Exam

15	Read §3.1: The Cantor Set 531: Does R(x) have the Cauchy Criterion? Prove your answer. Prove that the Cauchy Criterion and the Archimedean Property together imply the Least Upper Bound Property.

17	Read §3.2: Open and Closed Sets Exercises: 3.2.1, 3.2.3, 3.2.4

19	Reread §3.2: Open and Closed Sets Exercises: 3.2.5, 3.2.12, 3.2.13

22	Read §3.3: Compact Sets Exercises: 3.3.1, 3.3.2, 3.3.4, 3.3.5 531: Look up the definition of a connected set on p. 91. Use the Least Upper Bound property to prove that R is connected.

24	Reread §3.3: Compact Sets Exercises: 3.3.8, 3.3.9, 3.3.10

26	Read §4.1: Examples of Dirichlet and Thomae

29	Chapter 3 Exam

31	Read §4.2: Functional Limits Exercises: 4.2.1, 4.2.2, 4.2.4, 4.2.6

November (tentative)

Monday

Wednesday

Friday

2	Read §4.2: Functional Limits Exercises: 4.2.7, 4.2.8, 4.2.9

5	Read §4.3: Combinations of Continuous Functions Exercises: 4.3.2, 4.3.3, 4.3.7 531: Prove that if and ordered field is connected then the Least Upper Bound property holds.

7	Read §4.4: Continuous Functions on Compact Sets Exercises: 4.4.1, 4.4.2, 4.4.3, 4.4.5

9	Reread §4.4: Continuous Functions on Compact Sets Exercises: 4.4.6, 4.4.11, 4.4.12

12	Read §4.5: The Intermediate Value Theorem Exercises: 4.5.1, 4.5.5, 4.5.6 531: Prove that the Heine-Borel Theorem implies the Nested Interval Property.

14	Read §5.1: Are Derivatives Continuous?

16	Chapter 4 Exam

19	Read §5.2: Derivatives and the Intermediate Value Property Exercises: 5.2.1, 5.2.2, 5.2.3 531: 8.4.4, 8.4.5, 8.4.6, 8.4.7

Thanksgiving Break

26	Read §5.2: Derivatives and the Intermediate Value Property Exercises: 5.2.6 531: 8.4.8, 8.4.9

28	Read §5.3: The Mean Value Theorem Exercises: 5.3.4, 5.3.6 531: 4.3.6, 4.4.13

30	Reread §5.3: The Mean Value Theorem Exercises: 5.3.7, 5.3.11

December (tentative)

Monday

Wednesday

Friday

3	Read §5.4: A Continuous Nowhere-Differentiable Function Exercises: 5.4.1, 5.4.2, 5.4.3 531: 5.4.4, 5.4.5, 5.4.6

5	Chapter 5 Exam

Review

Final Exam: Tuesday, December 11, 1:30pm - 4:00pm