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MATH 74 HOMEWORK 5
Due Friday March 6th at 3:10pm. Throughout, let
X
and
Y
be metric spaces.
(1) Let
{
x
n
}
∞
n
=0
be a Cauchy sequence, and suppose that a subsequence
{
x
n
i
}
∞
i
=0
converges. Show that
the whole sequence
{
x
n
}
must converge.
(2) Show that
x
∈
A
if and only if inf
a
∈
A
d
(
x,a
) = 0.
First assume
x
∈
A
. Then
x
is a limit point of
A
and we’ve seen there is a sequence
{
a
n
}
∞
n
=0
with
a
i
∈
A
and lim
n
→∞
a
n
=
x
. Thus, for any
± >
0 there is an
a
n
such that
d
(
x,a
n
)
< ±
. This shows
inf
a
∈
A
d
(
x,a
) = 0 since if it were greater than zero, there would be some
±
for which there is no
a
with
d
(
a,x
)
< ±
.
Now assume inf
d
(
x,a
) = 0. Then for any
± >
0 there is an
a
∈
A
with
d
(
x,a
)
< ±
. So consider
points
a
n
gotten by choosing
±
= 1
/n
. This gives a sequence with lim
n
→∞
a
n
=
x
, so that
x
is a
limit point of
A
. But then
x
∈
A
.
(3) Show that if
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This note was uploaded on 05/04/2011 for the course MATH 73 taught by Professor Danberwick during the Spring '09 term at University of California, Berkeley.
 Spring '09
 DANBERWICK
 Math, Division

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