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 that the limit may exist even if f (x) does not.
Soluti
Math 320 Assignment 1, out of ?
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The natural numbers N = cfw_1, 2, . . . have properties that we will use without proof because I think you
learned them in Math 220. (1) Every subset of N has a greatest lower bound. (2)
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, there exists N such that |xn x| < .
But, using the reverse
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 in Rn is called convex if for every pair of points x a
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 you use.
(a) If x + y = 0, then y = x (i.e., every x in F
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 |x c|
whenever x B(c; ), x = c.
(a) Show that a funct
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 between two given sequences.
We will use vector notation to d
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 (c)| M |x c|
whenever x B(c; ), x = c.
(a) Show that a
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
Solution.
= :
Assume that f is continuous at p X. Let
> 0. T
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. Prove that if f : X Y is continuous on X then f E f (E) for
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 is an open
covering of (0, 1): let x (0, 1). Then, there
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 between two given sequences.
We will use vector notation to d
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 term you
leave out in this very special case.
Solution. We
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 = (bn bn+1 )1/2 . Show that bn (b1 b2 )1/3 as
2
n .
Sol
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
leave out in this very special case.
2. Let
an be a conver
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) If cfw_an is a convergent sequence in R, then lim an+1
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 )1/2 . Show that bn (b1 b2 )1/3 as
2
n .
2. Let , R an
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 = 3 k. So
3 k 3 k = 2 2 3 n n = k 3 k = 2 2 n n
and 3 mus
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 nonempty subsets S1 and S2 of R that are bounded from above
Math 320 Homework 3 Solutions
1. Chapter 2, Problem 5 in Rudin.
Solution. Let E = cfw_1/n : n Z+ cfw_1/n + 1 : n Z+ cfw_1/n + 2 : n Z+ . Then (i) E B(0; 4),
thus E is bounded. (ii) 0,1, and 2 are limit points of E. (Justify this as an exercise.) (iii) E
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 are allowed. Marks depend on quality of proofs. You can
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, 2013
Math 320
Name:
Page 2 of 9
1. Let L be a real number
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 are allowed. Marks depend on quality of proofs. You can
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, 2013
Math 320
Name:
Page 2 of 10
for class scores curved b
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 general false. The book of course has it right with
f
Math 320 Assignment 10
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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 is a continuous mapping of I
into I. Prove that f (x) =
Math 320 Assignment 8, out of 25
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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 convergence R of each of the following power series:
(a) (2
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 on the nal was 96. In computing your class mark I gave
Math 320 Assignment 09, out of 25
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In proofs it is a good habit to say when a hypothesis is being used.
1. (5 points) 4 #3. Let f be a continuous real function on a metric space X. Let Z(f ) be the set of all
p X at whic
Math 320 Assignment 7, out of 25
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Notation. Sequences cfw_an are in Rk or C. Make sure you are always clear about whether statements refer
to a sequence or a series.
1. 3 #6. Investigate the behaviour (convergence or di
MATHEMATICS 320, FALL 2016, PROBLEM SET 5
Solutions
1. (14 marks) If E is a subset of a metric space X, define the boundary
of E, E, by
E = cfw_x X : r > 0, Nr (x) E 6= and Nr (x) E c 6= .
(a) (3 marks) Prove that E = E E .
x E
r > 0 Nr (x) E 6= , and r
Math 320
Solutions to Assignment #1
September 18, 2015
Total marks = [25].
[2]
1. Let = max A. By definition, is an UB (Upper Bound) for A. If u is any UB for A, then
u since A. Hence is the least upper bound of A.
0
0
0
0
00
00
0
00
0
[2]
00
2. Assume (x
MATHEMATICS 320, FALL 2016, PROBLEM SET 2
SOLUTIONS
1. We will take for granted that 2, 3, 6 are irrational. The proof for 6 is identical
to # 3 on Assignment 1.
(a) [8] Let F = cfw_a + b 3 : a, b Q. Prove that F is a field.
The axioms (A2), (A3), (M2),
MATHEMATICS 320, SOLUTIONS TO PROBLEM SET 7
1. Let X = cfw_(x, y) R2 : (x, y) 6= (0, 0). Define f : X R by f (x, y) =
xy
. Prove that f is continuous on X but that it is not possible to
x2 +y 2
define f (0, 0) to make f continuous on R2 .
Solution. Suppos
MATHEMATICS 320, FALL 2016, PROBLEM SET 4
Due on Monday, October 31, in class
Write clearly and legibly, in complete sentences. You must provide complete explanations for all your solutions; answers without justification, even
if correct, will not be mark