(k) For any given sequence
(
s
n
),
(
s
n
)
has a convergent subsequence
(
s
n
k
).
4
(l) For any bounded sequence
(
s
n
),
lim sup
s
n
= sup
{
s
n
}
.
(m) It
α
is a subsequential limit of a bounded sequence
(
s
n
),
then
α
≤
lim sup
s
n
(n) If every subsequence of a sequence
(
s
n
)
is convergent, then
(
s
n
)
itself
must be convergent.
5
(o) If
(
s
n
)
is a divergent sequence, then some subsequence of
(
s
n
)
must
diverge.
(p) If
(
s
n
)
is unbounded above, then
(
s
n
)
has an increasing subsequence
(
s
n
k
)
which diverges to +
∞
.
6
2. Let
(
s
n
)
be a positive sequence such that
lim
n
→∞
s
n
+1
s
n
=
L >
1
.
Prove that
s
n
→
+
∞
.
Hint:
lim
n
→∞
x
n
= +
∞
for any number
x
such that
x >
1.
7
3.
Limits:
Determine the convergence or divergence of each of the following
sequences. If the limit exists, state what it is. (Hint: Use the previous problem,
and Theorem 3 from Section 17.)
(a)
s
n
=
n
3
3
n
(b)
s
n
=
n
!
n
10
(c)
s
n
=
n
n
n
!
8
4. For each of the following sequences, determine: (1) Whether the sequence con
verges; if it does, give the limit. (2) If the sequence does not converge, find the
set of subsequential limits (if any) and give
lim sup
and
lim inf.
(a)
s
n
=
1
n
+ 1
,
n
odd
3
2
n

1
,
n
even
(b)
s
n
=
1
n
,
n
= 3
,
6
,
9
,
· · ·
n
n
+ 1
,
n
= 1
,
4
,
7
,
10
,
· · ·
n
2
,
n
= 2
,
5
,
8
,
11
,
· · ·
(c)
s
k
=
k
sin
2
kπ
4
(d)
s
n
=
1
2
n
+ cos
nπ
6
9
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 Fall '08
 Staff
 Math, Mathematical analysis, Limit of a sequence, Limit superior and limit inferior, subsequence, Cauchy subsequence