Homework 5  Math 139
Due Oct 6th
Instructor
Mauro Maggioni
Office
293 Physics Bldg.
Office hours
Monday 1:30pm3:30pm.
Web page
www.math.duke.edu/˜mauro/teaching.html
Reading
: from Reed’s textbook: Sections 6.1,3.2
Problems
:
§
2.4: #10
§
2.6: #1, 3, 9
§
6.1: #1(a,c), 8
Additional Problem:
1. Let
{
a
n
}
and
{
b
n
}
be Cauchy sequences. Let
{
a
n
} ∼ {
b
n
}
mean that
a
n

b
n
→
0
1
.
Prove that
∼
is an equivalence relation:
{
a
n
} ∼ {
a
n
}
; if
{
a
n
} ∼ {
b
n
}
then
{
b
n
} ∼ {
a
n
}
;
if
{
a
n
} ∼ {
b
n
}
and
{
b
n
} ∼ {
c
n
}
, then
{
a
n
} ∼ {
c
n
}
.
2. Prove that the sum and product of Cauchy sequences is Cauchy.
3.
Let [
a
n
] denote the equivalence class of the Cauchy sequence
{
a
n
}
.
Given Cauchy
sequences
{
a
n
}
and
{
b
n
}
, define the sum and product of the equivalence classes containing
them by
[
a
n
] + [
b
n
] := [
a
n
+
b
n
]
[
a
n
]
·
[
b
n
] := [
a
n
b
n
]
Prove that these rules are welldefined by showing that if
{
a
n
} ∼ {
a
′
n
}
and
{
b
n
} ∼ {
b
′
n
}
,
then
{
a
n
+
b
n
} ∼ {
a
′
n
+
b
′
n
}
and
{
a
n
b
n
} ∼ {
a
′
n
b
′
n
}
4. If
C
denotes the set of equivalence classes of Cauchy sequences, then with the sum
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 Fall '08
 Pardon,W
 Math, Equivalence relation, Cauchy sequence, Cauchy, Cauchy sequences

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