MATH 425a ASSIGNMENT 1
FALL 2011 Prof. Alexander
Due Friday September 2.
Rudin Chapter 1 #1, 2, 4, 5 plus the problems (I) - (V) below:
(I) Take the set of all real numbers (not complex!) as the universe. Explain why each of the
following is true or false
MATH 425a
SAMPLE MIDTERM EXAM 1
Fall 2011
Prof. Alexander
(1)(a) State what it means for a point p to be a limit point of a set E .
(b) State the converse of the following (dont worry about whether the converse is true,
just state it): If x is a limit poi
MATH 425a SAMPLE FINAL EXAM SOLUTIONS
FALL 2011 Prof. Alexander
(1) Let > 0. There exists N such that n N = |fn (t) f (t)| for all t [a, b]. Then
for all x [a, b] we have
x
x
fn (t) dt
n N = |Fn (x) F (x)| =
a
x
f (t) dt
a
x
|fn (t) f (t)| dt
a
dt
a
(
MATH 425a
SAMPLE MIDTERM EXAM 1 SOLUTIONS
Fall 2011
Prof. Alexander
(1)(a) Every neighborhood of p contains a point x E with x = p.
(b) If x is not in the interior of E c then x is a limit point of E .
(c) (See text)
(2) Let x R and > 0. Since Q is dense
MATH 425a SAMPLE MIDTERM 2 SOLUTIONS
FALL 2011 Prof. Alexander
(1)(a)
an converges if the partial sums sn =
(b) See text.
n
j =1
form a converging sequence.
(2)(a)
(1)n
1.5n
n1.5
1/n
=
1.5
(n1/n )1.5
1.5 as n ,
so R = 1/1.5 = 2/3.
(b) Since (2/3)n 0, the
MATH 425a
SAMPLE MIDTERM EXAM 2
Fall 2011
Prof. Alexander
This is probably a little too long to realistically complete in an hour (especially with
#5a), but the problems are typical of what could be on an exam.
(1)(a) Give the denition of
an converges.
(
MATH 425a ASSIGNMENT 10
FALL 2011 Prof. Alexander
Due Wed. December 7, noon.
Rudin Chapter 7 #1, 2, plus the problems (A)(C) below, are to be turned in. Chapter 2
#3, 4, 8, 9 are for practice for the exam. You can turn it in by sliding it under my oce
doo
MATH 425a ASSIGNMENT 10
FALL 2011 Prof. Alexander
Due Monday November 28.
Rudin Chapter 5 #26, Chapter 6 #1, 2, 3ab, 8, plus the problems (I)-(V) below. Problems
1, 2, and (V) should be relatively quick ones, the type that most often appears on exams.
(I)
MATH 425a ASSIGNMENT 9
FALL 2011 Prof. Alexander
Due Wednesday November 16.
Rudin Chapter 5 #1, 2, 6, 7, 12, 13ab plus the problems (A)(E) below:
(A) A function f on R is called even if f (x) = f (x) for all x. Show that if f is even, and
dierentiable at
MATH 425a ASSIGNMENT 8
FALL 2011 Prof. Alexander
Due Monday November 7.
The due date is after the midterm, but this material IS covered on the midterm.
Rudin Chapter 4 #11 (rst sentence only), 12, 18, plus the problems (A)(G) below:
(A) Suppose f : [a, b]
MATH 425a ASSIGNMENT 7
FALL 2011 Prof. Alexander
Due Wednesday October 26.
Rudin Chapter 3 #11bc, Chapter 4 #2, 4, plus the problems (I)(VI) below:
(I) If
an converges, show that
Test wont work.)
an
n
converges. (Do not assume an 0. The Comparison
(II) Pr
MATH 425a ASSIGNMENT 6
FALL 2011 Prof. Alexander
Due Wednesday October 19.
Rudin Chapter 3 #7, 10, 20, 21 plus the problems (A)(F) below:
(A) Suppose an > 0 and
n
an converges. Show that
n
a2 converges.
n
(B) Let cfw_an be the Fibonacci sequence 1, 1, 2,
MATH 425a ASSIGNMENT 5
FALL 2011 Prof. Alexander
Due Wednesday October 12.
Rudin Chapter 2 #19, Chapter 3 #1, 3, 5, plus the problems (I)(VIII) below:
(I) Suppose sn 2, and tn 3 for all n.
(a) If snk + tnk c for a subsequence cfw_sn + tn , show that tnk c
MATH 425a ASSIGNMENT 4
FALL 2011 Prof. Alexander
Due Monday October 3.
Note this assignment is due after Midterm 1, but the material IS covered on the midterm.
Rudin Chapter 2 #12, 14, 16, 22 plus the problems (A)(E) below:
(A)(i) Show that in any metric
MATH 425a ASSIGNMENT 3
FALL 2011 Prof. Alexander
Due Wednesday September 21.
Rudin Chapter 2 #5, 7b, 8, 9ac, 11 (omit d5 ) plus the problems (I) - (V) below:
(I) Suppose A1 , . . . An are subsets of a metric space.
(a) If x is not a limit point of any of
MATH 425a ASSIGNMENT 2
FALL 2011 Prof. Alexander
Due Monday September 12.
Normally this would be due on a Friday, but it is Monday this time since Im out of town.
Rudin Chapter 1 #12, 13, 17, Chapter 2 #2, 3, 4 plus the problems (A) - (C) below:
(A) Let A