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18.100B,
Fall
2002,
Homework
4,
Solutions
Was
due
by
Noon,
Tuesday
October
1.
Rudin:
(1)
Chapter
2,
Problem
22
Let
Q
k
⊂
R
k
be
the
subset
of
points
with
rational
coeﬃcients.
This
is
countable,
as
the
Cartesian
product
of
a
ﬁnite
number
of
countable
sets.
Suppose
that
x
=(
x
1
,...,x
k
)
∈
R
k
.
By
the
density
of
the
rationals
in
the
real
numbers,
given
±>
0
there
exists
y
i
∈
Q
k
such
that

x
i
−
y
i

<±/k
,
i
=1
,...,k.
Thus
if
y
=(
y
1
,y
2
,...,y
k
)then
√
k

x
−
y
≤
k
max
i
=1

x
i
−
y
i

<±
shows
the
density
of
Q
k
in
R
k
.
Thus
R
k
is
separable.
(2)
Chapter
2,
Problem
23
Given
a
separable
metric
space
X,
let
Y
⊂
X
be
a
countable
dense
subset.
The
product
A
=
Y
×
Q
is
countable.
Let
{
U
a
}
,a
∈
A,
be
the
collection
of
open
balls
with
center
from
Y
and
rational
radius.
If
V
⊂
X
is
open
then
for
each
point
p
∈
V
there
exists
r>
0
such
that
B
(
p, r
)
⊂
V.
By
the
density
of
Q
in
X
there
exists
y
∈
Q
such
that
p
∈
B
(
y, r/
2)
.
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This note was uploaded on 12/16/2011 for the course STAT 5446 taught by Professor Frade during the Fall '09 term at FSU.
 Fall '09
 Frade

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