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ECON 831
Solutions Homework #2
Exercise 1:
(i) Consider
S
⊆
X
:
S
open in
X
⇔ ∀
x
∈
S,
∃
² >
0
/ N
²,X
(
x
)
⊆
S
.
(ii) Consider
(
x
m
)
∈
X
∞
:
x
m
→
x,x
∈
X
⇔ ∀
² >
0
,
∃
M
²
∈
R
/ d
(
x
m
,x
)
< ²
∀
m
≥
M
²
.
Exercise 2:
We consider 2 equivalent metrics
d
1
and
d
2
on
X
.
(i) We start with "
⇒
":
Consider a sequence
{
x
n
}
that converges in
(
X,d
1
)
towards
x
∈
X
: hence by de nition,
d
1
(
x,
n
,x
)
n
→∞
→
0
. One needs to show that
{
x
n
}
converges towards
x
(the same x!) in
(
X,d
2
)
: that is we need to show
d
2
(
x
n
,x
)
n
→∞
→
0
.
By assumption,
d
1
and
d
2
are equivalent, hence one has:
0
≤
d
2
(
x
n
,x
)
≤
bd
1
(
x
n
,x
)
∀
n
When
n
goes to in nity, the RHS goes to 0: hence one can conclude that
d
2
(
x
n
,x
)
→
0
.
"
⇐
":
Similar because the roles of
d
1
and
d
2
can be inverted.
(ii) We start with "
⇒
":
Consider an open subset
S
⊆
X
under
d
1
. By de nition, for any
x
∈
S
, there exists
² >
0
such that
N
²,
(
X,d
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 Fall '09
 Antoine
 Economics

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