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
3
Labor Day

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

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.

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

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

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



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

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.

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

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

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

Chapter 5 Exam
7  
Review               

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


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