Quiz 1
Name:
(25 points)
Let A and B be non-empty sets of real numbers. Suppose for every a A and every b B
we have a b. Prove that sup A inf B.
Proof.
Fix an arbitrary b B.
Then a A we have a b, that is, b is an upper bound of A. Then b sup A, the
least
Math 447, Fall 2016
Homework Assignment 6
Due at the beginning of class on Friday, October 14th. To get
full credit please explain your solutions clearly and legibly.
Problem 1. 17.2
Problem 2. 17.8
Problem 3. Let f and g be functions defined on R. Assume
Math 447
Solutions for homework 7
Problem 1. Assume that f is continuous on [0, ) and limx f (x) = 0,
i.e.,
> 0, M > 0, x > M, |f (x)| < .
Prove that f is uniformly continuous on [0, ).
Solution. Let > 0 be given. Since limx f (x) = 0, there exists N
suc
Math 447, Fall 2016
Homework Assignment 7
Due at the beginning of class on Friday, October 21th. To get full
credit please explain your solutions clearly and legibly.
Problem 1. Assume that f is continuous on [0, ) and limx f (x) = 0,i.e.,
> 0, M > 0, x
Math 447
Solutions for Homework 8
Problem 1. Let (S, d) be a metric space. Assume that f : S R and g : S R are uniformly
continuous on S. Prove that h : S R2 defined by h(s) = (f (s), g(s) is uniformly continuous
on S. Here R2 has the Euclidean metric.
So
Math 447, Fall 2016
Homework Assignment 8
Due at the beginning of class on Friday, October 28th. To get full
credit please explain your solutions clearly and legibly.
Problem 1. Let (S, d) be a metric space. Assume that f : S R and
g : S R are uniformly c
Math 447
Solutions for homework 6
Problem 1. 17.2
Solution. All of these functions are defined on R.
f , f + g and g f are discontinuous at 0, otherwise they are continuous.
g and f g are continuous on R.
f g(x) = 4 is continuous on R. The details are lef
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Math 447, Fall 2016
Problems for week 10
Please solve the following problems before the midterm exam.
Dont return your solutions.
Problem 1. 28.2 c), d).
Problem 2. 28.8
Problem 3. 29.4
Problem 4. 29.10
Problem 5. 29.18
Problem 6. Assume that f : R R is d
List of theorems for the second midterm exam, Math 447
At least one of the questions below will appear in the exam. Other questions
will be similar to the problems that appeared in the lecture notes and homework assignments.
1 Prove that (0, 1) is not a c
Math 447
Homework 11
Due Thursday, Dec. 1 2011, before class
1. Study Sections 20, 23, 24, 25 (without theorem 25.2) and Theorem
26.1.
2. Exercises 20.1, 20.5, 20.12, 20.13, 20.18 from the textbook;
3. Exercise 22.3, 22.4, 22.5, 22.13 from the textbook;
4
Math 447
Homework 6
Due Thursday, Oct. 13 2011, before class
Problem I (Holder and Minkowski Inequalities) This exercise will help you
prove two very important inequalities in analysis, the second one being
essential in showing that dp , 1 p < are metrics
Math 447
Homework 4
Due Sept. 22, 2011, before class
1. Exercises: 9.9 - 9.11 at p. 53 in the textbook.
2. Exercises: 9.12 - 9.16 at pp. 53-54 in the textbook.
3. Exercises: 10.1, 10.6, 10.7, 10.10 at pp. 62-63 in the textbook.
Math 447
Homework 10 and Review for Midterm
Due Thursday, Nov. 10 2011, before class
I Series.
1. Define series. Define convergent, absolutely convergent and divergent to series.
2. State the Cauchy Criterion for series and show that a series is
convergen
Math 447
Homework 5 and Review for Midterm
Due Thursday, Sept 29 2011, before class
I Fields.
1. State the axioms of a field. Show that in a field F :
2. (a) = a for any a F and (a1 )1 = a for any a F cfw_0;
3. 0 a = 0 and (1) a = a for any a F;
4. a b =
Math 447
Homework 12 and Review for Final
No due date
In addition to the topics in the previous reviews, see hw 5 and 10, the following topics will appear in the final exam.
I Continuity, Limits and Connectedness.
1. Define uniform continuity for a functi
Math 447
Homework 8
Due Thursday, Oct. 27 2011, before class
1. Exercise 14.1 (a), (b), (e), (f); 14.2 (f); 14.3 (e) from the textbook;
2. Exercises 14.5, 14.6 (a), 14.7, 14.9, 14.13 and 14.14 from the textbook.
3. Exercises 15.1 - 4 from the textbook.
Math 447
Homework 2
Due Sept. 8, 2011, before class
1. Prove that any finite set of real numbers has a maximum.
2. Show that the set S = cfw_r Q | r2 2, is a subset of Q (this should
be obvious), bounded from above by elements in Q which has no least
uppe
Math 447
Homework 9
Due Thursday, Nov. 3 2011, before class
1. Exercises 17.5, 17.6, 17.8, 17.10 (a, c), 17.12 and 17.13 from the textbook.
2. Exercises 21.4, 21.10 and 21.11 from the textbook.
3. Let (S1 , d1 ), (S2 , d2 ) and (S3 , d3 ) be metric spaces
Math 447
Homework 3
Due Sept. 15, 2011, before class
1. Exercises: 5.1, 5.2, 5.4, 5.6 at p. 28 in the textbook.
2. Exercises: 8.1, 8.4, 8.9 at pp. 42-43 in the textbook.
3. Exercise 8.5, at p. 42 in the textbook.
4. Exercises: Exercises: 9.1 and 9.6 at pp
Math 447
Homework 1
Due Sept. 1, 2011, before class
1. Consider the set M = cfw_0, 1, 2 with the following operations:
+
0
1
2
0
0
1
2
1
1
2
0
2
2
0
1
0
1
2
0
0
0
0
1
0
1
2
2
0
2
1
(a) Show that (M, +, ) is a field.
(b) Show that there is no order relatio
Math 447
Homework 7
Due Thursday, Oct. 20 2011, before class
1. Exercise 13.7 from the textbook.
2. Let (S, d) be a metric space and E S. The interior E o of E is defined
by:
E o = cfw_s E | r > 0 : Br (s) E
Show that E o is open and that the set E is ope