MATH 431/531 - Basic Analysis I
Fall 2012

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
 3 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
 7 Review

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

Home | Syllabus | Homework | Announcements | Study Guides | News