HOMEWORK 2 FOR MATH 521, FALL 2013
DUE MONDAY, 9 SEPTEMBER AT THE BEGINNING OF
LECTURE.
Do the following exercises from Abbott: 1.3.5, 1.3.7, 1.3.8, 1.4.4, 1.4.5.
1
Math 521 - Advanced Calculus I (Metcalfe)
Spring 2015
January 29, 2015
Quiz 1 Solutions
1. [10 points] Suppose that A, B R are nonempty and bounded above. Show that A B is
bounded above and that
sup(A B) = supcfw_sup A, sup B.
Since A, B R are bounded abo
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe
Due: February 3, 2015
Assignment 5
1. [10 points] If f and f g are one-to-one, does it follow that g is one-to-one? Justify your
answer.
2. [10 points] Prove that the collection F(N) of all nite subse
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe
Due: February 5, 2015
Assignment 6
1. [10 points] A complex number z is said to be algebraic if there are integers a0 , . . . , an , not all
zero, such that
a0 z n + a1 z n1 + + an1 z + an = 0.
Prove
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe
Due: February 26, 2015
Assignment 10
1. [10 points] Let E be the set of all limit points of a set E. Prove that E is closed. Do E and
E always have the same limit points?
2. [10 points] Let E denote t
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe
Due: February 19, 2015
Assignment 9
1. [10 points] Show that if (xn ) is unbounded, then there exists a subsequence (xnk ) such that
lim(1/xnk ) = 0.
2. [10 points] Let (xn ) be a bounded sequence of
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe
Due: February 10, 2015
Assignment 7
1. [10 points] Verify, using the denition of convergence of a sequence, that the following sequences
converge to the proposed limit:
1
(a) lim 6n2 +1 = 0.
(b) lim 3
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe
Due: January 15, 2015
Assignment 1
1. [10 points] Prove that n3 + (n + 1)3 + (n + 2)3 is divisible by 9 for all n N.
2. [10 points] Use the triangle inequality to establish that
|a| |b| |a b|,
for any
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe
Due: February 12, 2015
Assignment 8
1. [10 points] Suppose that cfw_xn is a convergent sequence and cfw_yn is such that for any > 0
there exists M such that |xn yn | < for all n M . Does it follow t
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe
Due: January 22, 2015
Assignment 3
1. [10 points] If sup A < sup B, then show that there exists an element b B that is an upper
bound for A.
2. [10 points] Use the Archimedean Property of R to rigorou
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe
Due: January 20, 2015
Assignment 2
These statements are implicitly claiming that the quantities in the conclusion exist. Be sure to
fully justify this as part of your proof.
1. [10 points] Let A be a
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe
Due: January 27, 2015
Assignment 4
1. [10 points] Suppose that S is a nonempty set of real numbers that is bounded. Prove that
inf S sup S.
2. [10 points] Dene S := cfw_x R : x2 < x. Prove that sup S
Math 521 - Advanced Calculus I (Metcalfe)
Fall 2012
September 28, 2012
Exam 1
Please show all of your work, and justify your answers completely.
Only one attempt at a solution will be graded per problem. If multiple approaches are provided,
it is your res
Math 521 - Advanced Calculus I (Metcalfe)
Fall 2012
September 28, 2012
Exam 1
Please show all of your work, and justify your answers completely.
Only one attempt at a solution will be graded per problem. If multiple approaches are provided,
it is your res
Math 521 - Advanced Calculus I (Metcalfe)
Spring 2015
January 29, 2015
Quiz 1
Please show all of your work, and justify your answers completely.
Only one attempt at a solution will be graded per problem. If multiple approaches are provided,
it is your res
HOMEWORK 5 FOR MATH 521, FALL 2013
DUE MONDAY, 30 SEPTEMBER AT THE BEGINNING OF
LECTURE.
Do the following exercises from Abbott: 2.3.3, 2.3.4, 2.3.7, 2.3.11, 2.3.12
1
HOMEWORK 6 FOR MATH 521, FALL 2013
DUE MONDAY, 7 OCTOBER AT THE BEGINNING OF
LECTURE.
Do the following exercises from Abbott: 2.4.2, 2.4.5, 2.5.1, 2.5.3, 2.5.6
1
HOMEWORK 4 FOR MATH 521, FALL 2013
DUE MONDAY, 23 SEPTEMBER AT THE BEGINNING OF
LECTURE.
Do the following exercises from Abbott: 2.2.1, 2.2.4, 2.2.6, 2.3.2, 2.3.5
Additional exercise(s) to turn in:
HW4.1:
(Re)-read Theorem 2.3.3, especially the proof of p
HOMEWORK 11 FOR MATH 521, FALL 2013
DUE WEDNESDAY, 4 SEPTEMBER AT THE BEGINNING OF
LECTURE.
Do the following exercises from Abbott: 1.2.1, 1.2.2, 1.2.4, 1.2.5, 1.2.9.
Additional exercise(s):
HW1.1:
Let A1 , A2 and B be sets with Aj B for j = 1, 2. For any
HOMEWORK 3 FOR MATH 521, FALL 2013
DUE MONDAY, 16 SEPTEMBER AT THE BEGINNING OF
LECTURE.
Do the following exercises from Abbott: 1.4.7, 1.4.9, 1.4.12, 1.5.9(a,b)
Additional exercise(s) to turn in:
HW3.1:
Prove that Q is countable and R is uncountable usin
HOMEWORK 7 FOR MATH 521, FALL 2013
DUE WEDNESDAY, 23 OCTOBER AT THE BEGINNING OF
LECTURE.
Do the following exercises from Abbott: 2.6.1, 2.6.2, 2.6.4, 2.7.3, 2.7.4, 2.7.5,
2.7.9
1
HOMEWORK 9 FOR MATH 521, FALL 2013
DUE MONDAY, 4 NOVEMBER AT THE BEGINNING OF
LECTURE.
Do the following exercises from Abbott: 3.3.4, 3.3.5, 3.3.8, 4.2.1, 4.2.2 Additional
exercise(s) to turn in:
In class we were proving one direction of Theorem 3.3.8, na
QUIZ 1
PRINTED Name:
(1) Let inf denote the greatest lower bound of a set of reals. Prove the following statement: Let A R be bounded
below, and let s R be a lower bound. Then s = inf(A) if and only if > 0, there exists an element a A satisfying
a < s + .
HOMEWORK 8 FOR MATH 521, FALL 2013
DUE MONDAY, 28 OCTOBER AT THE BEGINNING OF
LECTURE.
Do the following exercises from Abbott: 3.2.3, 3.2.5, 3.2.7, 3.2.9 Additional
exercise(s) to turn in:
HW8.1:
Write out in your own words a proof that Q is neither open
SYLLABUS FOR MATH 521-002, FALL 2013
1. Course Information
Instructor: Hans Christianson
Time and day: MWF, 12:00 - 12:50 pm.
Location: Phillips 385
Oce hours: Wednesday 2-3 pm and Friday 1-2 pm, or by appointment
Contact Information:
Oce: Phillips 304B
MATH 1242 PRACTICE FOR TEST #2.
This practice sheet covers sections 6.3-6.6 and 7.1-7.5 (old book sections 5.7-5.10 and
6.1-6.3 and 6.5). The problems on this sheet are intended to help you study for the test,
but you are also responsible for questions li