Introductory Analysis 1Fall 2009
Homework 7Due Monday, November 23, 2009
1. Rosenlicht, Chapter IV, Problem 34 (p. 94).
Solution.
a) fn : [0, 1] R is dened by fn (x) =
1, 2, . . . Is cfw_fn uniformly convergent?
x
for n =
1 + nx2
To solve this well use o

Introductory Analysis 1Fall 2009
Exam 2October 30, 2009
ANSWERS
1. (30 points) Let X, Y be metric spaces and let f : X Y . Suppose there
is a constant C 0 (that is, a non-negative real number C ) such that
dY (f (p), f (q ) C dX (p, q )
for all p, q X . P

Introductory Analysis 1Fall 2009
Final Exam
Due: Wednesday, December 9, 2009
Note on what can be accepted: In solving these problems you can use any
theorem in the textbook or seen in class.
1. For each k in R, let ak R, ak 0. If S is a nite set of positi

Introductory Analysis 1Fall 2009
Final Exam with hints
Due: Wednesday, December 9, 2009
Note on what can be accepted: In solving these problems you can use any
theorem in the textbook or seen in class.
1. For each k in R, let ak R, ak 0. If S is a nite se

Introductory Analysis 1Fall 2007
Homework #3Solutions
Some sayings to live by, or to ignore. Mathematics is a very powerful but
delicate tool. If you use it wrong, it breaks easy.
Mathematics can be very treacherous. Once you start wading through its
dark

Introductory Analysis 1Fall 2009
Homework 4Solutions to Exercises 13
Note: This homework consists of a lot of very simple exercises, things you
should do on your own. A minimum part of it will be due Monday, October 5,
2009.
1. Let X be a set and let d1 ,

Introductory Analysis 1Fall 2009
Homework 5Solutions
1. Rosenlicht, Chapter IV, #3
Solution.
Ill present two proofs.
Proof 1. Let F be a closed subset of E . Then
f 1 (F ) = f 1 (F ) S1 f 1 (F ) S2 = (f |S1 (F )1 (f |S2 (F )1 .
Recall the result seen in c

Introductory Analysis 1Fall 2009
Homework 6Solutions to Exercises 1-5
1. Let X be a metric space and consider the set cfw_0, 1 as a metric space with
the discrete metric.
(In other words, d is dened in cfw_0, 1 by d(0, 0) = d(1, 1) = 0; d(0, 1) =
d(1, 0)

Introductory Analysis 1Fall 2007
Exam 1Solutions
1. Dene the following concepts:
(a) Compact metric space.
Solution. A metric space is compact i every open covering has
a nite subcovering.
(b) Cauchy sequence.
Solution. A sequence of points cfw_pn in a m