McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 6: Solutions
1. Let (xn ) be a sequence. A point x R is called an accumulation point of (xn ) if th
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 1: Solutions
1. Let f : D E be a function and let A D, B E. Prove the following:
(a) f (f 1 (B) B.
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
Assignment 9
This is a practice assignment and it will not be marked or collected. However, you
should careful
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 10: Solutions
1. Let I be a closed and bounded interval and let f : I R be a continuous function su
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 8: Solutions
1. Using the - denition of the limit of a function, prove that
x
a
=
for all a R, a =
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 9
You should carefully work out all problems. However, you only have to hand in solutions to
proble
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 7
You should carefully work out all problems. However, you only have to hand in solutions to
proble
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 5: Solutions
1. If (bn ) is a bounded sequence and lim (an ) = 0, prove that lim (an bn ) = 0.
Solu
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 3: Solutions
1. Let A and B be two nonempty subsets of R. Prove that A B is bounded above if and on
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 2: Solutions
1. Let (G, +) be a group. Prove directly from the group axioms that for all a, b, c G
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 4
You should carefully work out all problems. However, you only have to hand in solutions
to proble
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 10
This is a practice assignment and it will not be marked or collected. However, you
should carefu
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 8
You should carefully work out all problems. However, you only have to hand in solutions to
proble
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 7: Solutions
1. Let x1 R \ cfw_0 and let
xn+1 = xn +
1
xn
n N
(a) Prove that lim (xn ) = + if x1 >
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
Assignment 2: Solutions
1. Let A be a countably infinite set and let B A. Prove that B is countable.
Solution:
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
Assignment 4: Solutions
1. Let S := cfw_
1
1
: n, m N. Determine sup S and inf S.
n m
Solution:
For all n, m N
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
Assignment 7: Solutions
1. Show directly from the definition that the following sequences are Cauchy sequences
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
Assignment 3: Solutions
1. Let A R, B R be two nonempty sets bounded from above. The sum of A and B is the set
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
Assignment 5: Solutions
1. Use the definition of the limit of a sequence to show that:
2
n
1
n 1
=0
(b) lim
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
Assignment 8: Solutions
1. Let x1 R \ cfw_0 and let
xn+1 = xn +
1
xn
n N
(a) Prove that lim (xn ) = + if x1 >
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
Solutions for the Sample Final Exam Questions
1. For any nonempty set A R we define A := cfw_a : a A. Suppose
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 5
You should carefully work out all problems. However, you only have to hand in solutions
to proble
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 3
You should carefully work out all problems. However, you only have to hand in solutions
to proble
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 2
You should carefully work out all problems. However, you only have to hand in solutions
to proble
Class Notes for MATH 255.
by
S. W. Drury
c 2006, by S. W. Drury.
Copyright
Contents
0 LimSup and LimInf
1
1 Metric Spaces and Analysis in Several Variables
1.1 Metric Spaces . . . . . . . . . . . . .
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
Assignment 10
This is a practice assignment and it will not be marked or collected. However, you
should carefu
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
Assignment 10: Solutions
1. Let I = [a, b] and let f : I R be continuous on I. Assume that f (a) < 0 and that
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
Assignment 8
You should carefully work out all problems. However, you only have to hand in solutions to
proble
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
Assignment 3
You should carefully work out all problems. However, you only have to hand in solutions
to proble