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
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 Spring '08
 FLANAGAN
 Work, Metric space, Rational number, Cauchy sequence, Xn

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