HOMEWORK 6, M600
University of Delaware
1. If E X and if f is a function dened on X , the restriction of f to E is the function
g whose domain of denition is E , such that g (p) = f (p) for p E . Dene f1 and f2
on R2 by
xy 2
(x, y ) = (0, 0)
x2 +y4
f1 (x
Supplements to the Exercises in Chapters 1-7 of Walter Rudins
Principles of Mathematical Analysis, Third Edition
by George M. Bergman
This packet contains both additional exercises relating to the material in Chapters 1-7 of Rudin, and
information on Rudi
University of Delaware
Department of Mathematical Sciences
Math 600
C. Bacuta
Review for Mid II
Know:
1. Denitions, Theorems, Propositions, Examples as presented in class, or the corresponding
versions from Rudin PMA book. The topics that will be consider
University of Delaware
Department of Mathematical Sciences
Math 600
C. Bacuta
Review for the Final Exam
December 6 2013, 3:30-5:30 PM sharp, in Sharp Lab 100
Know:
1. Denitions, Theorems, Propositions, Examples as presented in class, or the corresponding
HOMEWORK 5 M600
University of Delaware
1. Determine whether the following series are convergent or divergent giving arguments
to support your claim.
n=1
(d)
n=1
(g )
1
(a)
(b)
1
n1+ n
n=1
(1+1/n)n
n
(e)
n=1
(n!)2
(2n)!
(c)
1+2n +5n
3n
(f )
n=1
n=1
n2 4
(n
Math 600 - Homework 1, due Wednesday, September 11, 2013
You may discuss all the problems with your classmates or with me and no one else.
However, the nal answers must be in your own words.
Please do not copy solutions from books or from notes of stude
HOMEWORK 4 M600
University of Delaware
1. Given two sequences cfw_xn and cfw_yn in a metric space (X, d) such that lim xn = x
n
and lim yn = y . Prove that lim d(xn , yn ) = d(x, y ).
n
n
2. Consider the sequence dened recursively as
s1 =
2,
sn+1 =
2+
s
HOMEWORK 3 M600
University of Delaware
Denition : Let A be a non-empty subset of a metric space X and p an element of X .
Dene the distance from p to A as
d(p, A) = inf cfw_ d(p, a) : a A
Note that d(p, A) is the inmum of a non-empty subset of R which is
HOMEWORK 2 M600
Please check, the solutions to Homework #1. Good short solutions are expected.
1. Prove that the set of all irrational numbers is not countable.
2. Let f : S T be a function. Prove that the following statements are equivalent.
(a) f is one
University of Delaware
Department of Mathematical Sciences
Math 600
C. Bacuta
Review for Mid I
Know:
1. Denitions, Theorems, Propositions, etc., as presented in class, or the corresponding versions from Rudin PMA book -Up to 3.12, page 55. Do not miss:
(1