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IE 500 – Fall 2009
E. Alper Yıldırım
HOMEWORK 5 (due Wednesday, December 9 in class)
1. Let (
x
n
) be a sequence of real numbers. Let
∞
<
lim inf
n
→∞
x
n
=
a <
lim sup
n
→∞
x
n
=
b <
+
∞
.
(a) For each real number
± >
0, prove that there exist inﬁnitely many elements of the sequence (
x
n
)
such that
x
n
∈
(
a

±,a
+
±
).
(b) For each real number
± >
0, prove that there exist inﬁnitely many elements of the sequence (
x
n
)
such that
x
n
∈
(
b

±,b
+
±
).
2. Using the result from the ﬁrst problem, prove that there exist two subsequences (
a
n
) and (
b
n
) of the
sequence (
x
n
) such that lim
n
→∞
a
n
=
a
and lim
n
→∞
b
n
=
b
.
3. Let (
x
n
) be a sequence of real numbers such that
∞
<
lim inf
n
→∞
x
n
=
a
≤
lim sup
n
→∞
x
n
=
b <
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This note was uploaded on 04/15/2010 for the course INDUSTRIAL ie513 taught by Professor Zeynephuygur during the Spring '10 term at Bilkent University.
 Spring '10
 zeynephuygur

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