1.207
The present value of the
𝑛
payments is the
𝑛
th partial sum of the geometric
series
𝑥
+
𝛽𝑥
+
𝛽
2
𝑥
+
𝛽
3
𝑥
+
. . .
which (using (1.31)) is given by
Present value =
𝑠
𝑛
=
𝑥
−
𝛽
𝑛
𝑥
1
−
𝛽
1.208
By Exercise 1.93, there exists an open set
𝑇
⊇
𝑆
1
such that
𝑇
∩
𝑆
2
=
∅
. For every
x
∈
𝑆
1
, there exists an open ball
𝐵
(
x
) such that
𝐵
(
x
)
⊆
𝑇
and therefore
𝐵
(
x
)
∩
𝑆
2
=
∅
.
The collection
{
𝐵
(
x
)
}
of open balls is an open cover for
𝑆
1
. Since
𝑆
1
is compact there
exists a finite subcover, that is there exists points
x
1
,
x
2
, . . . ,
x
𝑛
in
𝑆
1
such that
𝑆
1
⊆
𝑛
∪
𝑖
=1
𝐵
(
x
𝑖
)
Furthermore, for every
x
𝑖
, there exists
𝑟
𝑛
such that
𝐵
(
x
𝑖
) =
x
𝑖
+
𝑟
𝑛
𝐵
where
𝐵
is the unit ball. Let
𝑟
= min
𝑟
𝑛
.
𝑈
=
𝑟𝐵
is the required neighborhood.
1.209
Clearly
𝑋
×
𝑌
is a normed linear space. To show that it is complete, let (
z
𝑛
) be
a Cauchy sequence in
𝑋
×
𝑌
where
z
𝑛
= (
x
𝑛
,
y
𝑛
). For every
𝜖 >
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 Fall '10
 Dr.DuMond
 Macroeconomics, Topology, Topological space, normed linear space, Foundations of Mathematical Economics, ????th partial sum

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