MATH 255
Assignment 3
Solutions
1. We have
n
lim
n
k=1
1
k2
= lim
3 + k3
n n
n
= lim
n
1
=
0
=
=
1
n
n
k=1
nk 2
+ k3
n3
k
n
n
k=1
2
3
k
n
1+
x2
dx
1 + x3
1
ln(1 + x3 )
3
1
0
ln 2
3
x2
is continuous and from the proof
1 + x3
of the fact that a continuous f
MATH 255 ASSIGNMENT 3 This assignment is due in class on Monday, January 31 Problems Please justify carefully your answers. 1. [10 points] Is it true that in any metric space (X, d), cl(D(x, r) = cfw_y X : d(x, y ) r. Provide a proof or nd a counterexampl
MATH 255 ASSIGNMENT 3, short solutions Problems Please justify carefully your answers. 1. [10 points] Is it true that in any metric space (X, d), cl(D(x, r) = cfw_y X : d(x, y ) r. Provide a proof or nd a counterexample. Solution. Counterexample. Let X be
MATH 255 ASSIGNMENT 2, short solutions 1. [10 points] Let X be the collection of all sequences of positive integers. If x = (nj ) j =1 and y = (mj ) are two elements of X , set j =1 k (x, y ) = inf cfw_ j : nj = mj and d(x, y ) = 0
1 k(x,y )
if x = y if
McGill University
Department of Mathematics and Statistics
MATH 255 Analysis 2, Winter 2016
Assignment 3
This assignment is due Monday, March 7, at the end of the class in class. Late
assignments will not be accepted.
1. For x = (x1 , , xn ) Rn , set kxk
McGill University
Department of Mathematics and Statistics
MATH 255 Analysis 2, Winter 2016
Assignment 1
This assignment is due Monday, February 1, at the end of the class in class. Late
assignments will not be accepted.
1. Find all the points at which th
MATH 255
ASSIGNMENT 7
This assignment is due in class on Tuesday, April 14
Please make a photocopy of your assignment before submitting it
Problems
Please justify carefully your answers.
1. [10 points] (1) Let f R() on [a, b]. Let g be a bounded function
MATH 255
ASSIGNMENT 6
This assignment is due in class on Wednesday, April 1
Problems
Please justify carefully your answers.
1. [10 points] Let (xn )
n=1 be a sequence of strictly positive numbers. Prove the
following:
(1)
xn+1
lim inf
lim inf n xn .
n
n
MATH 255
ASSIGNMENT 2
This assignment is due in class on Monday, January 26
Problems
Please justify carefully your answers.
1. [10 points] Let X be the collection of all sequences of positive integers. If x = (nj )
j=1
are
two
elements
of
X,
set
and y = (
MATH 255
ASSIGNMENT 3
This assignment is due in class on Wednesday, February 11
Problems
Please justify carefully your answers.
1. [10 points] Is it true that in any metric space (X, d),
cl(D(x, r) = cfw_y X : d(x, y) r.
Provide a proof or find a countere
MATH 255
ASSIGNMENT 1
This assignment is due in class on Monday, January 19
Problems
Please justify carefully your answers.
1. [10 points] Let (xn ) be a bounded sequence of real numbers. Prove that
lim inf xn = sup cfw_t : cfw_n : xn < t is finite .
n
2.
MATH 255
ASSIGNMENT 4
This assignment is due in class on Friday, February 20
Problems
Please justify carefully your answers.
1. [10 points] Let X be the collection of all sequences of positive integers. If x = (nj )
j=1
are
two
elements
of
X,
set
and y =
McGill University
Department of Mathematics and Statistics
MATH 255 Analysis 2, Winter 2016
Assignment 2
This assignment is due Monday, February 15, at the end of the class in class.
Late assignments will not be accepted.
1. Let cfw_Ki iI be a collection
MATH 255 Hon. Analysis II
Prof. V. Jaksic
Solutions for Assignment I
1. Let c R. By density of the rationals in R, there is a sequence cfw_qn nN of rationals such that
limn qn = c. By density of the irrationals in R, there is a sequence cfw_bn nN of irrat
MATH 255 Hon. Analysis II
Solutions for Assignment IV
1. Let x cl(S). By our previous characterization of closedness, for each n N, B(x, 1/n) S is
nonempty, so we can pick xn there. This defines a sequence (xn )n in S with d(xn , x) < 1/n.
On the other ha
MATH 255 Hon. Analysis II
Solutions for Assignment III
1. For x = (x1 , . . . , xn ) Rn , set kxk = max |xk |.
(a) It is clear that kxk 0 with k0k = 0.
Moreover, if kxk = 0, then for each 1 6= j 6= n, |xj | max1kn |xk | = 0. We conclude
x = 0.
Let R, then
McGill University
Department of Mathematics and Statistics
MATH 255 Honours Analysis 2, Winter 2016
Solutions 13 by Eric Hanson
Assignment 2
Problem 1. Let cfw_Ki iI be a collection of compact sets in R such that for any finite subcollection
cfw_Ki1 , Ki2
Analysis II Tutorial
Solution sketches
Alexandre Tomberg
January 27, 2011
1.1
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1
Let X be a set and A, B X.
1. Prove that (A B) A B.
2. Find an example of X, A, B s.t. (A B) 6= A B.
3. Find a
McGill University
Department of Mathematics and Statistics
MATH 255 Analysis 2, Winter 2016
Assignment 4
This assignment is due Monday, March 28, at the end of the class in class. Late
assignments will not be accepted.
The problems 12-16 are bonus problem
MATH 255
ASSIGNMENT 5
This assignment is due in class on Wednesday, March 11
Problems
Please justify carefully your answers.
1. [10 points] Let (X, dX ) and (Y, dY ) be metric spaces and f : X Y . Prove that
the following statements are equivalent.
(1) f
FACULTY OF SCIENCE
FINAL EXAMINATION
MATHEMATICS MATH 255
Honours Analysis 2
Examiner: Professor S. W. Drury
Date: Wednesday, 22 April 2009
Associate Examiner: Professor V. Jaksic
Time: 2: 00 pm. 5: 00 pm.
INSTRUCTIONS
Answer all questions in the booklets
MATH 255
ASSIGNMENT 5, short solutions
1. [10 points] Let (X, dX ) and (Y, dY ) be metric spaces and f : X Y . Prove that
the following statements are equivalent.
(1) f is continuous on X.
(2) For any open set V Y , the set
f 1 (V ) = cfw_x X : f (x) V ,
Analysis III, Assignment 6
Nicolas Resch
December 13, 2013
[1] Proof. Fix a = (an )
n=1 in X =
Q
n=1 Xn .
For a given x = (xn )
n=1 , define
x(k) = (a1 , a2 , . . . , ak , xk+1 , xk+2 , . . .).
Let f : X cfw_0, 1 be a continuous function. For 1 j k, defin
Analysis III, Assignment 2
Nicolas Resch
December 13, 2013
1. Proof. This statement is false. Consider the following counter-example. Let X = R, x = 0, r = 1,
and d : R R R be a discrete metric, i.e. for x, y R,
(
d(x, y) =
1, x 6= y
0 x = y.
Then, B(0, 1
Analysis III: Assignment 1
Nicolas Resch
1. Let x, y, z X.
i. d(x, y) 0. This is clear, as if x = y, then d(x, y) = 0 by definition, and if x 6= y, then k(x, y) > 0
1
(as the number at which they first differ must be a positive integer), and so d(x, y) =
MATH 354
ASSIGNMENT 1
This assignment is due in class on Tuesday, September 17
Problems
Please justify carefully your answers.
1. [10 points] Let X be the collection of all sequences of positive integers. If x = (nj )
j=1
are
two
elements
of
X,
set
and y
Analysis III, Assignment 5
Nicolas Resch
Friday, November 8, 2013
1. Proof. We begin by extending f to a continuous function on all of [1, 1] by letting
x 1, 21
0
f(x) = f (x) x 12 , 12
0
x 12 , 1 .
Since f was continuous on [ 12 , 12 ] and f ( 21 ) = f (
Math 354
Assignment 6
Please make a photocopy of this assignment before submitting it. The
graded assignments will be returned during the review session in December.
This assignment is due on Fri, Nov 29, and solutions will discussed in the
tutorial on th
6. Proof. Let (fn )
n=1 be a Cauchy sequence in (X, k k). Then, given > 0, there exists N > 0 such
that for n, m N ,
kfn fm k = max (|(fn fm )(t)| + |(fn fm )0 (t)|) < .
t[a,b]
Fix x [a, b]. Then, for n, m N ,
0
|fn (x) fm (x)| |fn (x) fm (x)| + |fn0 (x)
Math 354
Assignment 5
[1] [20 points] Let f be a continuous function on [ 12 , 12 ] with f ( 12 ) = f ( 21 ) =
0. Let sk be a sequence of continuous functions on [1, 1] such that:
(1) sRk (x) 0 for x [1, 1];
1
(2) 1 sk (x)dx = 1;
(3) For any 0 < < 1,
Z
l
Analysis III, Assignment 4
Nicolas Resch
December 13, 2013
1. Proof.
First of all, observe that the function f is uniformly continuous. Indeed, given > 0,
choosing = gives that for all x, y satisfying d(x, y) < = , d(f (x), f (y) = d(x, y) < . Hence,
its
Math 354
Practice problems for the final exam
If you wish to improve your homework score, you can submit the solutions
of this problem set on Dec 13, at the beginning of the review session. Each
problem is worth 20 points. You can improve your score up to
MATH 354
ASSIGNMENT 3
This assignment is due in class on Tuesday, October 15
Problems
Please justify carefully your answers.
1. [10 points] Let X be the collection of all sequences of positive integers. If x = (nj )
j=1
are
two
elements
of
X,
set
and y =
Math 354
Assignment 4
Due in class on Thursday, October 24
[1] [15 points] Let (X, d) be a compact metric space and let f : X 7 X be
a function such that d(x, y) = d(f (x), f (y) for all x, y X. Show that f is
onto.
[2] [10 points] Let (X, d) be a metric
MATH 354
ASSIGNMENT 2
This assignment is due in class on Friday September 27
Problems
Please justify carefully your answers.
1. [10 points] Is it true that in any metric space (X, d),
cl(D(x, r) = cfw_y X : d(x, y) r.
Provide a proof or find a counterexam