Supplements to the Exercises in Chapters 1-7 of Walter Rudins
Principles of Mathematical Analysis, Third Edition
by George M. Bergman
This packet contains both additional exercises relating to the material in Chapters 1-7 of Rudin, and
information on Rudi
MAT 337 Homework assignment 3.
Due Wednesday, March 4, 2015
Each problem is worth 10 point
Problem 1. Determine whether the following series converge or diverge.
3n
(a) n3 +1
n=2
n!
(b) nn
n=1
(c) (log1n)n
n=2
1
(d) (1)n (e n 1)
n=1
(e) sin( n )
n=1
4
Pro
MAT337 Winter 2016 Assignment 2
February 5, 2016
You should submit the assignment during class on Friday 12/02.
1. In a metric space (X, d), let K, L X be compact. Prove that K L, K L are also
compact.
2. In the standard Euclidean space R, find the interi
MAT337 Winter 2016 Assignment 4
March 4, 2016
You should submit the assignment during class on Friday 11/03.
1. Find all intervals I R on which the sequence fn (x) =
uniformly.
x2n
,
n+x2n
fn C(I) converges
x+n
2. Find all intervals I [0, ) on which the s
MAT337 Winter 2016 Assignment 5
March 18, 2016
You should submit the assignment during class on Wednesday 30/03.
1. Let K, L C[0, 1] be two compact subsets. Define K + L := cfw_f (x) + g(x) : f K, g
L. Prove or give a counter-example:
(a) K + L is closed
MAT337 Winter 2016 Assignment 3
February 19, 2016
You should submit the assignment during class on Friday 26/02.
In the following, the Cantor set C [0, 1] is a metric spaces with the standard metric inherited from [0, 1].
1. If A, B X are connected, and A
MAT 337
Sample Midterm Exam 2
NAME
NO AIDS ALLOWED
Total: 250 points, not including a bonus problem
Problem 1 [30 points]
(a) Give an example of a function f : lR lR that is continuous at every
irrational point, and that is discontinuous at every rational
Item ID: 11332
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Faculty of Arts and Sciences
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Sample Final Exam, April-May 2014
MAT 337 H1
Intro Real Analysis
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Instructor: Regina Rotman
Duration - 3 hours
No aids allowed
Total marks for this paper is 400
Please wr
MAT337, Midterm 1
Sample
w Answer
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.
Note: This is a sample of answers(or ideas) to your first midterm exam.
there are several correct answers, but I will only write one.
Problem 1. [20 points]
Determine which of the following sequences converge:
1
1
(a
Exercises for Section 1.3
1. Prove the converse of the Generalized Heine-Borel: A compact subset
of Rn is closed and bounded.
Proof. Let A be a compact set in Rn . To see that A is bounded,
consider the open cover
O = cfw_B (0, r) : r R ,
n
O
of all open
DA
300
18
[976
Supp.
VlATH
Solutions Manual to Walter.
Rudins Principles 0f
Mathematical Analysis
Roger Cooke, University of Vermont mmmwnmnn-Wuommwww-w
WWWM
Chapter 6 V'
The RiemannStieltjes
Integral,
Exercise 6.1 Suppose oz increases on [(1, b], a g :30
2A
300
28
l 976
Supp.
VIATH
Solutions Manual to Walter
Rudin’s Principles- 0f
Mathematical Analysis
Roger Cooke, University of Vermont Chapter 4
Continuity ‘
Exercise 4.1 Suppose f is a real function deﬁned on R1 which satisﬁes
,gim mm + h) — f<m — h>1= 0
DA
300
?8
I 976
Supp.
WATH
Solutions rManual to Walter
Rudins Principles 0f
Mathematical Analysis
Roger Cooke, University of Vermont Chapter 3
Numerical Sequences and
Series
Exercise 3.1 Prove that convergence of {57,} implies convergence of Is
the conve
2A
300
28
1976
Supp.
VIATH
Solutions Manual-to Walter
* Rudins Principles 0f
Mathematical Analysis
Roger Cooke, University of Vermont Chapter 2
Basic TopolOgy
Exercise 2.1 Prove that the empty set is a subset of every set.
Solution. Let Z denote the empty
3A
300
28
i 976
Supp.
VIATH
Solutions Manual to Walter
Rudins Principles 0f
Mathematical Analysis
Roger Cooke, University of Vermont Chapter 1
The Real and Complex
Number Systems
Exercise 1.1 If 7" is rational (7' 7E 0) and :2: is irrational, prove that 7
MAT 337
Sample Midterm Exam
NAME
NO AIDS ALLOWED
Total: 250 points, not including a bonus problem
Problem 1 [20 points]
(a)Determine whether the following sequence (an ) , where an = (1
n=1
1
1
).(1 n2 ) converges.
32
1
)(1
22
(b) Let (xn ) be Cauchy wit
MAT 337 H1S Introduction to Real Analysis
Books:
Principles of Mathematical Analysis, by W. Rudin (3rd edition)
Supplementary Books:
Mathematical Analysis, a straightforward approach, by K. G. Binmore
Real Analysis and Applications, by K. Davidson and A.
2A
300
L976
supp.
VIATH
Solutions Manual to Walter
Rudins Principles 0f
Mathematical Analysis
Roger Cooke, University of Vermont Chapter 7
Sequences and Series of
Functions
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Exercise 7.1 Prove that every uniformly con