Introductory Analysis 2Spring 2010
Homework 3Solutions
1. Problem 15 of Chapter VII, Rosenlicht (p. 162)
Solution. There is a question on how to interpret
n,m=1 an bm . I
gave you a lot of latitude, but it all depends what one wants to do with
it. Lets su

Introductory Analysis 2Spring 2010
Final ExamSolutions
May 5, 2010
INSTRUCTIONS: Each exercise is worth 10 points. If an exercise has two or more parts, each part is worth the same
fraction of 10 points. The maximum grade you can get for this test is 100

Introductory Analysis 2Spring 2010
Exam 2
April 2, 2010
1. (10 points) We saw in class that a subset A of R is Lebesgue measurable if and only if for every > 0 there exists
a open set U such that A U and m(U \A) < , also if and only if there exists a clos

Introductory Analysis 2Spring 2010
Exam 1-February 22, 2010
SOLUTIONS
1. Assume that the power series
its radius of convergence.
n=0
an (z 5)n converges for z = 7, but diverges for z = 3. Determine, with proof,
Solution. Let r be the radius of convergence

Introductory Analysis 2Spring 2010
Homework 10Solutions
Exercise 1 f : R [0, ] is measurable.
Solution. It should be observed that f is ONLY dened for x E . So writing f = E f is wrong. It isnt terribly
wrong, but it just isnt right. But a direct proof is

Introductory Analysis 2Spring 2010
Homework 9
Due April 7, 2010
Wrapping up Chapter 3
1. Recall the exercise from Exam 2. Let be a function from open subintervals of R to [0, ] so that if I = (a, b),
a b, then (I ) [0, ). Assume it sends the empty interv

Introductory Analysis 2Spring 2010
Homework 7
Due: Wednesday, March 17, 2010, St. Patricks Day
SOLUTIONS
Due: Wednesday, March 17, St. Patricks Day
1. Recall the denition of an outer measure (See Homework 6). Recall also what was said in class: The constr

Introductory Analysis 2Spring 2010
Homework 6Due: Wednesday, March 3, 2010
Solutions
Im only adding solutions in cases where I felt most proofs that I saw where not quite satisfactory. There is a typo in
Exercise 1, part f. The corrected version has been

Introductory Analysis 2Spring 2010
Homework 5Solutions
With improved, wider margins!
Note: This homework assumes that you know what you know from Calculus.
1. Let cfw_fn be an equicontinuous sequence of real valued functions dened on the compact interval

Introductory Analysis 2Spring 2010
Homework 4. Solutions
1. Let
series
n=0 an be a convergent series of complex numbers
n
n=0 an z has radius of convergence r 1.
Solution.
(or real numbers; that doesnt matter). Prove: The power
This is a totally silly exe

Introductory Analysis 2Spring 2010
Homework 1Solutions
Exercise 1 Assume z, w C, w = z and |z | = 1. Prove
zw
= 1.
1zw
Solution.
A quick way of doing this is to use that |z | = |z | = z z = 1, thus
zw
zw
z (z w )
zz zw
1 zw
= |z |
=
=
=
= |1| = 1.
1zw
1zw