640:411
FALL 2012
SOME HOMEWORK SOLUTIONS: ASSIGNMENT 7
Chapter 4: 2. Since f (E ) is closed and f is continuous, f 1 f (E ) is closed. But E
f 1 f (E ) f 1 f (E ) , so E f 1 f (E ) , and f (E ) f (E ).
To show that the inclusion may be proper, take Y =
www.myengg.com
CET PHYSICS 2014
VERSION CODE: B 1
A person is driving a vehicle at uniform speed of 5 ms-1 on a level curved track of radius 5 m.
The coefficient of static friction between tyres and road is 0.1. Will the person slip while
taking the turn
640:411
Fall 2012
Assignment 1
Due Monday, September 17
Exercises:
Chapter 1: 1, *2, 3, *5, *6, 7, *9
Remarks, hints, and extra questions:
6. Remember that as you approach problem 6 you know very little about fractional
exponents; in fact, only the inform
640:411
FALL 2012
SOME HOMEWORK SOLUTIONS: ASSIGNMENT 1
2. If we replace 2 by 3 throughout then the proof given in Example 1.1 applies almost
without change to show that there is no rational p with p2 = 12.
5. We rst show that sup(A) exists. Since A is no
640:411
FALL 2012
SOME HOMEWORK SOLUTIONS: ASSIGNMENT 2
13. From the triangle inequality, |x| = |(x y) + y| |x y| + |y|, so |x| |y| |x y|.
Similarly, |y| |x| |y x| = |x y|, and these two inequalities imply |x| |y| |x y|.
Of course, this has nothing to do
640:411
SUPPLEMENTARY PROBLEMS
FALL 2012
1. For r a rational number and a cut, prove that > r if and only if r .
2. Using the denitions of multiplication of cuts (real numbers) given in the text or
equivalently in class, prove the following statements. No
640:411
Fall 2012
Assignment 3
Turn in starred problems (including 3.A) Monday, Octovber 1
Exercises:
Chapter 2: *2, 5, *6, *7, 8, 9(a)(c), *(d), *(e), (f), 10
*3.A Two metrics d1 and d2 on the same set X are called equivalent if for every x X
(1)
(2)
(2)
640:411
FALL 2012
SOME HOMEWORK SOLUTIONS: ASSIGNMENT 3
2. Every integer is an algebraic number, since n Z is a root of the equation 1 z n = 0.
Thus the set of algebraic numbers is not nite.
For xed n 1 and xed (n + 1)-tuple of integers a = (a0 , . . . ,
640:411
Fall 2012
Assignment 4
Turn in starred problems Thursday, October 11. Note change of day!
Exercises:
Chapter 2: 13, *15, *17, *19, 20, 21, *22, 23, *25
Extra credit: 18
Remarks, hints, and extra questions:
1. Remember that we have an exam on Monda
640:411
FALL 2006
SOME HOMEWORK SOLUTIONS: ASSIGNMENT 4
1
1
1
+ 2 n, m N, m n . It is easy to
nN
n
nm
1
n N , which shows (i) that the set is closed
verify that the set of limit points is cfw_0
n
and hence compact, since it is bounded, and (ii) that its
640:411
Fall 2012
Assignment 5
Turn in starred problems Monday, October 29.
Exercises:
Chapter 3: 1, 2, *3, 4, *5, 21, *22
5.A* Let (sn )nN be a sequence in R. Show that
lim sup sn = inf sup sn
n
(4.1)
N 1 nN
(recall that the right hand side of (4.1) is s
640:411
FALL 2012
HOMEWORK SOLUTIONS: ASSIGNMENT 5
Rudin Chapter 3: 1. Proof sketch: if limn sn = s then from |sn | |s| |sn s|
(see Exercise 1.13) it follows easily that limn |sn | = |s|. However, if sn = (1)n then
limn |sn | = 1 but (sn ) is not converge
640:411
Fall 2012
Assignment 6 Revised
We have lost two days of class due to Hurricane Sandy. I still hope to have time at
the end of the course for a substantial discussion of integration, and with this in mind we
will omit some material from Chapter 3,
640:411
FALL 2012
SOME HOMEWORK SOLUTIONS: ASSIGNMENT 6
an is a telescoping sum, with
Rudin Chapter 3: 6. (a) If an = n + 1 n then
N
N
N + 1 1. Thus n=1 an is not bounded and the series diverges.
n=1 an =
(b) Here
n+1 n
1
1
.
=
n
2n n
n( n + 1 + n)
The se
AglaSem Admission
KCET MATHEMATICS 2014
VERSION CODE: C 2
1.
Which one of the following is not correct for the features of exponential function given by f
(x) = bx where b > 1?
(1) For very large negative values of x, the function is very close to 0.
(2)