Sometimes true:
Let
s
n
=
N
= (1
,
2
,
3
,
4
,
· · ·
). Every subsequence of
(
s
n
) is unbounded.
Let
s
n
=
⎧
⎨
⎩
1
,
n
odd
1
/n,
n
even
.
Then
(
s
2
n
−
1
) = (1
,
3
,
5
,
7
,
· · ·
)
is un-
bounded;
(
s
2
n
) = (1
/
2
,
1
/
4
,
1
/
6
,
1
/
8
,
· · ·
is bounded (and has limit 0).
(e) If (
s
n
) is a bounded, monotone sequence, then (
s
n
) is a Cauchy sequence.
Always true:
A bounded monotone sequence is convergent; a convergent
sequece is a Cauchy sequence.
2

(f) If (
s
n
) is a bounded sequence, then (
s
n
) has a Cauchy subsequence.
Always true:
This is Theorem 2, Section 19.
(g) Every oscillating sequence has a convergent subsequence.
Always true:
If (
s
n
) oscillates, then
α
= lim sup
s
n
̸
= lim inf
s
n
=
β
,
and there is a subsequence
s
n
k
which converges to
α
and a subsequence
s
n
p
which converges to
β
,
α
̸
=
β
.
(h) Every oscillating sequence diverges.
Always true:
There are subsequences which converge to two di
ff
erent
limits, as shown above.
3

(i) If (
s
n
) is an unbounded sequence, then either
s
n
→
+
∞
or
s
n
→ −∞
.
Sometimes true:
s
n
=
N
is an unbounded sequence and
s
n
→
+
∞
.
s
n
=
⎧
⎨
⎩
n,
n
odd
1
/n,
n
even
is unbounded, but
s
n
does not diverge to +
∞
or to
−∞
.
(j) If
(
s
n
)
is a bounded sequence and
α
= sup
{
s
n
}
,
then
(
s
n
)
has a
subsequence which converges to
α
.
Sometimes true:
If
(
s
n
)
is bounded and increasing, then
s
n
→
α
=
sup
{
s
n
}
Let
s
n
= (2
,
0
,
1
/
2
,
2
/
3
,
3
/
4
,
· · ·
,
n
−
1
n
,
· · ·
). Then
sup
{
s
n
}
= 2,
(
s
n
)
does not have a subsequence which converges to
2
(all subsequences
converge to
1).

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- Fall '08
- Staff
- Mathematical analysis, Limit of a sequence, Limit superior and limit inferior, Sn, subsequence