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Problem Set 2
Economics 204  August 2006
Due Tuesday, 8 August in Lecture
1. Theorem 4 from the lim inf/lim sup handout: Let
{
x
n
}
be a sequence of real numbers.
Then
lim
x
→∞
x
n
=
x
∈
R
∪ {∞
,
∞}
if and only if
lim inf
x
→∞
x
n
= lim sup
x
→∞
x
n
=
x.
Prove this theorem for the case that
x
is ﬁnite.
2. Using the deﬁnition of an open set, prove that (0
,
1) is an open subset of
R
(a) Under the usual absolute value metric.
(b) Under the discrete metric.
3. Construct a sequence of real numbers
(a) That is unbounded and has exactly three cluster points.
(b) That is bounded and has inﬁnitely many cluster points.
4. Prove that lim
n
→∞
(
√
n
2
+
n

n
) = 1
/
2.
5. Closure, Boundary, Interior, etc.
(a) Let
A
,
B
, denote subsets of a space
X
. Determine whether the following equations
hold; if an equality fails, determine whether one of the inclusions
⊆
or
⊇
holds.
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This note was uploaded on 03/06/2009 for the course ECON 0204 taught by Professor Staff during the Summer '08 term at University of California, Berkeley.
 Summer '08
 Staff
 Economics

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