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Kenneth A. Ross
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YSIS:
THE THEORY OF
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Undergraduate Texts in Mathematics
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Math 451
Written Homework Set 4
due on Tuesday, February 14
Problem 1. Let (sn )nN be a sequence of positive numbers which converges to 0. Show
that the sequence (tn )nN , with tn = sn for every n, al
Math 451
Written Homework Set 1
due on Tuesday, January 24
Problem 1. (Algebraic and Order Properties of R) Prove the following statements making
use of the first 14 axioms for R.
(a) If a < b and c 6
Math 451 Solutions to Homework Set 7
Problem 1. 1. We have
lim+
x0
whereas
lim
x0
x
x
= lim+ = lim+ 1 = 1,
x x0 x x0
x
x
= lim
= lim (1) = 1.
x x0 x x0+
x
x0 x
These also show that lim
does not
Math 451, Sections 1 and 2
Midterm Exam 1
Wednesday, 15 February 2017
Name:
This exam consists of 8 problems. Each problem or part of a multiquestion problem is
worth 5 points, and there are 85 point
Math 451
Alternate Midterm Exam 2
Thursday, 23 March 2017
Name:
The total number of points on this exam is 90. For maximum score, you need to get 75
points or more.
On the last page you will find a sm
Math 451
3rd Quiz
4/11/2017
Name:
For maximum score you need to do at least 2 out of the following 3 problems.
For Problem 3 you may want to use the statement of Problem 2 (not its proof).
Problem 1.
Math 451
Midterm Exam 2
Wednesday, 22 March 2017
Name:
The total number of points on this exam is 90. For maximum score, you need to get 75
points or more.
On the last page you will find a small list
Math 451 Solutions to Homework Set 1
Problem 1. We will use the numbering of the first 14 axioms for R given
in Ross book.
(a) We consider two cases:
(i) If c = d, then we have b + d = b + c, and ther
Math 451, Sections 1 and 2
Alternate Final Exam
Friday, 21 April 2017
Name:
The total number of points on this exam is 110. For maximum score, you need to get 90
points or more.
On the last page you w
Math 451
Alternate Midterm Exam 2
Thursday, 23 March 2017
Name:
The total number of points on this exam is 90. For maximum score, you need to get 75
points or more.
On the last page you will find a sm
Math 451 Solutions to Homework Set 6
1
= 0, we
n n
Problem 1. (a) Since sin(x) is continuous at 0, and since lim
must have that lim sin(1/n) = sin(0) = 0.
n
sin(x)
= 1, and since n1 nN is a sequence o
Math 451
Homework Set 9Review Homework
April 14, 2017
Note 1. Recall the definitions of finite, countably infinite, countable, uncountable set.
4
S
Problem 1. Let Ai , i = 1, 2, 3, 4, be sets, and ass
Math 451 Solutions to Homework Set 5
Problem 1. (a) The series diverges. We can use the Comparison Test: for
every n > 2, we have that n 1 > n/2, therefore, for every n > 2,
n1
n/2
1
.
> 2 =
2
n
n
2n
Math 451
Written Homework Set 5
due on Thursday, March 9
Problems 15. Do exercises 14.2, 14.4, 14.5 parts (a) and (b), 14.6 and 14.10 from
Chapter 2, Section 14 of Ross book (2nd edition).
P
P
Proble
Math 451
Solutions for the 1st Quiz
Problem 1. For each of the following sets, indicate whether it is bounded above and/or below,
and find its supremum and/or infimum, and its maximum and/or minimum w
Math 451 Solutions to Homework Set 3
Problem 1. (a) Assume that for some n N we have that P (n) is true,
that is, assume that n2 + 5n + 1 is an even integer. Then we have that
n2 + 5n + 1 = 2k for som
Math 451, Sections 1 and 2
Final Exam
Thursday, 20 April 2017
Name:
The total number of points on this exam is 110. For maximum score, you need to get 90
points or more.
On the last page you will find
Math 451 Partial solutions to Homework Set 9
Problem 1. Assume the negation of the conclusion, that all the Ai , i =
1, 2, 3, 4 are countable, that is, there exist injective functions fi : Ai N.
4
4
S
Math 451
Solutions for the 2nd Quiz
Problem 1. For each of the following series, determine whether it converges or diverges.
State clearly the method/s you are using, and briefly explain how you apply
Math 451
Written Homework Set 6
due on Tuesday, March 21
Practice problems (not to be turned in): Exercises 17.1, 17.2 and 17.3 from Chapter
3, Section 17 of Ross book (2nd edition).
X
1
sin
. Observe
Math 451
Written Homework Set 8
due on Thursday, April 13
Problem 1. Prove  cos(x) cos(y) 6 x y for all x, y R.
Problem 2. (a) Show sin(x) < x for all x > 0. [Hint. Consider first the function
f (
Math 451, Sections 1 and 2
Midterm Exam 1
Wednesday, 15 February 2017
Name:
This exam consists of 8 problems. Each problem or part of a multiquestion problem is
worth 5 points, and there are 85 point
Math 451 Solutions to Homework Set 8
Problem 1. Consider any two x, y R with x 6= y (say x < y). Then,
by the Mean Value Theorem (since cos(u) is continuous on [x, y], and its
derivative sin(u) exists
Math 451
Midterm Exam 2
Wednesday, 22 March 2017
Name:
The total number of points on this exam is 90. For maximum score, you need to get 75
points or more.
On the last page you will find a small list
Math 451 Solutions to Homework Set 2
Problem 1. Since A is countably infinite, we know that there is a bijection
f : A N. Similarly, since B is countably infinite, there is a bijection
g : B N. We cou
Math 451 Solutions to Homework Set 4
Problem 1. Consider > 0. We need to find N such that, if n > N , then
tn 0 = tn  < . Note that, by our assumption, tn  = tn = sn , thus
tn  < is equivalent
Math 451
Topics from chapter 3: Limits of functions and continuity
We consider only functions f : dom(f ) R where dom(f ) R.
(1) Continuous functions:
(a) A function f (x) is continuous at x0 dom(f )
Math 451
Handout on supremums, inmums and limits.
These are some properties of supremums and limit supremums of sequences. It is important that you know
these properties and how to prove them.
(1) sup
Math451 Mock Midterm I
Question 1
(a) (10%) Write down the formal definition of the statement an a.
Solution: For all > 0, there exists N N such that for all n N , an a < .
(b) (10%) For each series