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Homework 8. Solutions
December 1, 2008
12.2 Show: lim sup

s
n

= 0 if and only if lim
s
n
=0
Proof
Suppose lim sup

s
n

=0. Let
± >
0, so there is
N
such that
∀
k > N
implies sup
{
s
n

:
n > k
}
< ±
. This implies that
∀
n > N
we
have

s
n

< ±
, which is just the deﬁnition of lim
s
n
= 0.
Now suppose lim

s
n

= 0. Let
± >
0, there exist
N
so that
∀
n > N
implies

s
n

< ±
. So for all
k > N
sup
{
s
n

:
n > k
}
< ±
, which implies
that lim sup

s
n

= 0.
12.4 Show that: lim sup
s
n
+
t
n
≤
lim sup
s
n
+ lim sup
t
n
for bounded
sequences
s
n
and
t
n
. Give an example of two sequences for which the
equality does not hold.
Proof
Let (
s
n
) and (
t
n
) be bounded sequences. Then, the sets
{
s
n
:
n
∈
N
}
,
{
t
n
:
n
∈
N
}
and
{
s
j
+
t
k
:
j,k
∈
N
}
are bounded. Moreover,
for each
N
∈
N
the subsets
{
s
n
:
n > N
}
,
{
t
n
:
n > N
}
and
{
s
n
+
t
n
:
n > N
}
are bounded (and are nonempty), by Completeness Axiom the
subsets have a supremum. Let
s
N
=
sup
{
s
n
:
n > N
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