Math 337 Winter 2010, Problem set 2
Due Wed Feb. 9
It is recommended that you try all the problems. Hand in only those which are
starred for grading. The hints which are given are just suggestions. You may nd
a dierent or better way of doing the problem.
TEST 2 SOLUTIONS
(1) (a) Suppose E is a subset of a metric space (X, d). The diameter
of E is dened as diam E = supcfw_d(x, y ) : x, y E . Suppose X
is complete and (En ) is a decreasing sequence of closed non-empty
subsets of X such that limn diam En = 0
TEST 1 SOLUTIONS
[30] (1)
[15] (a) Suppose that f : R R. Give the denition of limxa f (x) =
L.
limxa f (x) = L means, by denition, that for every > 0, there
exists > 0 such that, for x R, if 0 < |x a| < then |f (x) f (a)| <
.
[15] (b) Prove directly from
Math 337 Winter 2010, Problem set 6 Solutions to selected problems
8.1 B. First, since ntn 0 for t [0, 1) we have fn (x) 0 for
all x [0, 1]. Calculus reveals that on [0, 1] fn has a maximum
at
n
2n n
1
and fn (xn ) =
(
xn =
).
2n + 1
2n + 1 2n + 1
1
Sin
Math 337 Winter 2010, Problem set 6
Not to be handed in
It is recommended that you try all the problems.
8.1 B, E, F, K, J
8.2 A, D, E, G, J In J replace [a, b] by any compact metric space X .
C (X, V ) stands for the set of all continuous functions f : X
PROBLEM SET 5: SOLUTIONS TO STARRED
PROBLEMS
(1) Show that if f is integrable on [a, b] then |f | is also and that
b
b
f a |f |.
a
Let f+ (x) = maxcfw_f (x), 0 and f (x) = maxcfw_f (x), 0. Then
f = f+ f= and |f | = f+ + f .
Claim. For any a s < t b, we ha
Math 337 Winter 2010, Problem set 4
Due Fri. Mar. 19
It is recommended that you try all the problems. Hand in only those which are
starred for grading. The hints which are given are just suggestions. You may nd
a dierent or better way of doing the problem
Math 337 Winter 2010, Problem set 5
Due Fri. Mar. 19
It is recommended that you try all the problems. Hand in only those which are
starred for grading. The hints which are given are just suggestions. You may nd
a dierent or better way of doing the problem
PROBLEM SET 4: SOLUTIONS TO STARRED
PROBLEMS
2.9H Show that |P (X )| = |X | for any set X .
Suppose, for a contradiction, that |P (X )| = |X |, so that there exists
a bijection f : X P (X ). Set
A = cfw_x X : x f (x) X.
Let us see that A is not in the ima
Math 337 Winter 2010, Problem set 3
Due Wed Feb. 24
It is recommended that you try all the problems. Hand in only those which are
starred for grading. The hints which are given are just suggestions. You may nd
a dierent or better way of doing the problem.
PROBLEM SET 3: SOLUTIONS TO STARRED
PROBLEMS
(1) Prove that a closed ball cfw_x : d(x, a) r is a closed set.
Let B = cfw_x : d(x, a) r. To show that B is closed, we check
that whenever (xn ) B is a convergent sequence, its limit x is in B .
Certainly, for
PROBLEM SET 2: SOLUTIONS TO STARRED
PROBLEMS
2.8G. Fill in the details of how the Completeness Theorem implies
the Least Upper Bound Principle.
Let S be a non-empty set of real numbers which is bounded below.
Consider the set of all integers which are low
PROBLEM SET 1: SOLUTIONS TO STARRED
PROBLEMS
2.4A. Compute the limit and, using = 106 , nd an integer N that
satises the limit denition.
(c) limn
3n
n!
We have
3n
3
3
3
=
0,
n!
n n1
1
as n . Let us label the terms of the sequence an , that is, an :=
Then
Math 337 Winter 2010, Problem set 1
Due Mon. Jan. 25
It is recommended that you try all the problems. Hand in only those which are
starred for grading. The hints which are given are just suggestions. You may nd
a dierent or better way of doing the problem
Problem Set 1
Monday, January 25, 2010
Section 2.3: A, D
Section 2.4: A(c)*(e), G, I, J*
Section 2.5: E, I*, J
Section 2.6: J, L*, N
Section 2.7: A, B, I*, L
A.
Show that (an)=(ncosn(n)/(n+2n)n=1 has a convergent subsequence.
B.
Does the sequence (bn)=(n+