Math 320 Homework 10
1. A function f : R R is said to satisfy a Lipschitz condition of order at c if there exists a
positive number M (which may depend on c) and a ball B(c; ) such that
|f (x) f (c)|
M ATH 140 B - HW 2 S OLUTIONS
Problem 1 (WR Ch 5 #11). Suppose f is dened in a neighborhood of x, and suppose
f (x) exists. Show that
lim
h0
f (x + h) + f (x h) 2 f (x)
= f (x).
h2
Show by an example
Math 320 Assignment 1, out of ?
send corrections to db5dmath.ubc.ca
The natural numbers N = cfw_1, 2, . . . have properties that we will use without proof because I think you
learned them in Math 220
Math 320 Homework 8
In the next three questions, we will investigate certain metric spaces that consist of certain sequences
in R. For this, we will dene dierent metrics to measure the distance betwee
Math 320 Homework 9
1. Dene the diameter of a subset S of metric space Y to be
diam S = supcfw_dY (p, q) : p, q S
Prove that f : X Y is continuous at p X if and only if
lim diam f B(p; ) = 0
0
2. Prov
Math 320 Homework 9
1. Dene the diameter of a subset S of metric space Y to be
diam S = supcfw_dY (p, q) : p, q S
Prove that f : X Y is continuous at p X if and only if
lim diam f B (p; 0) = 0
0
Solut
Math 320 Homework 10
1. A function f : RmapstoR is said to satisfy a Lipschitz condition of order at c if there exists
a positive number M (which may depend on c) and a ball B(c; ) such that
|f (x) f
Math 320 Homework 8
In the next three questions, we will investigate certain metric spaces that consist of certain sequences
in R. For this, we will dene dierent metrics to measure the distance betwee
Math 320 Homework 1
1. Prove there is no rational number q such that q 2 = 12.
2. Let F be an ordered eld. Show that the following hold (where x, y, z are in F . In
each case, indicate which axioms yo
Math 320 Homework 7 Solutions
1. Suppose that an+1 an and lim an = 0. Prove that
n
N
(1)n an
n=0
(1)n an aN +1
n=0
That is, the error introduced by truncating the series is no larger than the rst ter
Math 320 Homework 6 Solutions
1. (a) Dene the sequence cfw_an recursively by a1 = , a2 = and an+2 = 1 (an+1 + an ). Show
2
that cfw_an converges to 1 + 2 .
3
3
(b) Suppose that b1 0, b2 0, and bn+2
Math 320 Homework 7
1. Suppose that an+1 an and lim an = 0. Prove that
n
N
n
(1)n an aN +1
(1) an
n=0
n=0
That is, the error introduced by truncating the series is no larger than the rst term you
lea
Math 320 Homework 5
1. Prove that convergence of cfw_xn in R implies convergence of cfw_|xn | in R. Is the converse true?
2. True or False. If True give a proof. If False give a counter-example.
(a)
Math 320 Homework 6
1
1. (a) Dene the sequence cfw_an recursively by a1 = , a2 = and an+2 = 2 (an+1 + an ). Show
1
2
that cfw_an converges to 3 + 3 .
(b) Suppose that b1 0, b2 0, and bn+2 = (bn bn+1
Math 320 Homework 5 Solutions
1. Prove that convergence of cfw_xn in R implies convergence of cfw_|xn | in R. Is the converse true?
Solution. Suppose that cfw_xn converges to x. Then, given > 0, the
Math 320 Homework 4 Solutions
1. Give an example of an open cover of the interval (0, 1) which has no nite subcover.
Solution. Consider the collection F = cfw_(1/n, 1) : n Z+ . First, we show that F i
Math 320 Homework 4
1. Give an example of an open cover of the interval (0, 1) which has no nite subcover.
2. If E R, prove that the set of all isolated points of E is countable.
3. Prove that the set
Math 320, Fall 2007, Homework Set 11
(due on Friday November 30 2007)
Instructions
This is the last homework set of the semester. If you
need the homework assignment to study for the final,
you shoul
Math 320, Fall 2007, Homework Set 8
(due on Wednesday November 7 2007)
Instructions
Homework will be collected at the end of lecture on Wednesday.
You are encouraged to discuss homework problems amo
Math 320 Homework 3
1. Chapter 2, Problem 5 in Rudin.
2. Chapter 2, Problem 6 in Rudin.
3. Chapter 2, Problem 7 in Rudin.
4. Chapter 2, Problem 8 in Rudin.
5. Chapter 2, Problem 9 in Rudin.
6. A set E
Math 320 Homework 1
1. Prove there is no rational number q such that q 2 = 12.
Proof. Suppose that
m 2
n
= 12, with m, n Z+ having no common factors. Since
mm=223nn
3 must divide m exactly. Write m =
Math 320 Homework 2
1. Let S1 and S2 be nonempty subsets of R that are bounded from above.
(a) Prove that if S1 , S2 cfw_x R : x 0, then
supcfw_xy : x S1 , y S2 = (sup S1 )(sup S2 )
(b) Find two none
The University of British Columbia
Final Examination - December 7, 2012
Mathematics 320
Time: 2.5 hours
Last Name
First
Signature
Student Number
Special Instructions:
No books, notes, or calculators a
Math 320 Midterm Test, Nov. 15, 2013
Closed book exam. The test has 4 questions and is out of 40.
Last Name:
First Name:
Student Number:
Question Points Score
1
10
2
10
3
10
4
10
Total:
40
Nov. 15, 20
The University of British Columbia
Final Examination - December 7, 2012
Mathematics 320
Time: 2.5 hours
Last Name
First
Signature
Student Number
Special Instructions:
No books, notes, or calculators a
Math 320 Midterm Test, Oct 11, 2013
Closed book exam. The test has 4 questions and is out of 40.
Last Name:
First Name:
Student Number:
Question Points Score
1
10
2
10
3
10
4
10
Total:
40
Oct. 11, 201
Denition 0.1 For f : X Y and A X let
f (A) = cfw_y Y |y = f (x) for some x A,
which is a subset of Y . For B Y let
f 1 (B) = cfw_x X|f (x) B.
In class I think I said that
f 1 (f (A) = A
but this is in
Math 320 Assignment 10
send corrections to db5dmath.ubc.ca
This will not be collected. I will post solutions on line later.
1. 4, #14. Let I = [0, 1] be the closed unit interval in R. Suppose that f i
Math 320 Assignment 8, out of 25
send corrections to db5dmath.ubc.ca
This is the last assignment before Midterm on Friday Nov 15. This midterm will cover hw 5,6,7,8.
1. 3, #9. Find the radius of conve
Comments on Math 320 Final
Averages and histograms shows percentage marks y on the nal with 2 percentage points
added. Example: if you got 50 out of 80 on the nal then y = 50/80 100 + 2 65. The
top y
Math 320, Fall 2007, Homework Set 3
(due on Wednesday September 26 2007)
Instructions
Homework will be collected at the end of lecture on Wednesday.
You are encouraged to discuss homework problems a
Math 320, Fall 2007, Homework Set 1
(due on Wednesday September 12, 2007)
Instructions
Homework will be collected at the end of lecture on Wednesday.
You are encouraged to discuss homework problems
Math 320, Fall 2007, Homework Set 4
(due on Wednesday October 3 2007)
Instructions
Homework will be collected at the end of lecture on Wednesday.
You are encouraged to discuss homework problems amon
Math 320, Fall 2007, Homework Set 7
(due on Wednesday October 31 2007)
Instructions
Homework will be collected at the end of lecture on Wednesday.
You are encouraged to discuss homework problems amo
Math 320, Fall 2007, Homework Set 2
(due on Wednesday September 19 2007)
Instructions
Homework will be collected at the end of lecture on Wednesday.
You are encouraged to discuss homework problems a