This preview shows pages 1–2. Sign up to view the full content.
4130 HOMEWORK 3
Due Thursday February 18
(1) Let
{
x
n
}
and
{
y
n
}
be Cauchy sequences of rational numbers. Prove that
{
x
n
} ∼ {
y
n
}
if and only if for all
ε >
0 there exists
N
∈
N
such that for all
m,n > N
,

x
m

y
n

< ε
.
Let (
*
) be the condition:
for all
ε >
0 there exists
N
∈
N
such that for all
m,n > N
,

x
m

y
n

< ε
.
We need to prove that (
*
) holds if and only if:
for all
ε >
0 there exists
N
∈
N
such that for all
n > N
,

x
n

y
n

< ε
.
Clearly, (
*
) implies the second condition. So we just need to check that if
{
x
n
} ∼
{
y
n
}
then (
*
) holds. So suppose
{
x
n
} ∼ {
y
n
}
. Then let
ε >
0. Choose
N
1
∈
N
such that if
n > N
, then

x
n

y
n

< ε/
2. Since
{
y
n
}
is a Cauchy sequence, we may
also choose
N
2
∈
N
such that if
m,n > N
2
then

y
m

y
n

< ε/
2. Now suppose
m,n >
max
{
N
1
,N
2
}
. Then

x
m

y
n
 ≤ 
x
m

y
m

+

y
m

y
n

< ε
as required.
(2) (a) Using the formula for the partial sums of a geometric series, or otherwise, check
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 FLANAGAN
 Work

Click to edit the document details