Homework 5
Section 4.4
2a) False; By theorem 4.4.8, “
Every unbounded sequence contains a monotone subsequence that
has either
+∞ ?? − ∞
as a limit.”
Every sequence does not have a convergent subsequence.
2b) True; By theorem 4.4.7, “
Every bounded sequence has a convergent subsequence
.” So by
definitions 4.4.9, the subsequential limit must not be empty.
2c) False; By definition 4.4.9, if S
n
converges to s, then the lim inf S
n
= lim sup S
n
. This is true
only if s
∈ ℝ
2d) True
2e)False: S
n
=n for every natural number n. It is not bounded about but the lim Inf S
n
= 1.
3a) Since S
1
= 0, S
2
=2, S
3
=0, S
4
=2, and S
5
=0, then the subsequential limit is {0,2}. The Lim Sup
S
n
=2 and the Lim Inf S
n
=0
3b) t
n
= ( 0, ½, 2/3, ¼, 4/5, 1/6, 6/7, …)
The subsequential limit is {0,1},thus the Lim Sup t
n
=1
and the Lim Inf t
n
=0
3c) Since u
1
=-2, u
2
=0, u
3
=-18, u
4
=0, u
5
= -50, u
6
=0
The subsequential limit is {
−∞
, 0}. The
lim Sup u
n
=0 and the Lim Inf u
n
=
−∞
.
3d) Let V
n
=
𝑛𝑠𝑖𝑛(𝜋𝑛)
2
. Then V
1
= 1, V
2
=0, V
3
= -3, V
4
=0, V
5
=5, V
6
=0, and V
8
= -8. When n is even,
the sequence is 0, for all n
∈ ℕ
. When n is odd, the sequence diverges to
+∞ ?? − ∞
The
subsequential limit is {
+∞, 0, −∞
} with
+∞
as is the Lim Sup V
n
and
−∞
is the limit inf V
n.
4a) Since w
1
= -1, w
2
=1/2, w
3
= -1/3, w
4
= ¼, then w
n
converges to 0
∀? ∈ ℕ
lim sup w
n
= lim
inf w
n
= 0
4b) The subsequence can be written as X
3n-2
=0
∀?
,
X
3n-1
=1
∀?
, and X
3n
converges to
+∞
The subsequential limit is {0, 1,
+∞}
and the Lim Sup X
n
=
+∞
; Lim Inf X
n
=
0
4c) The subsequence can be written as Y
2n
=6n
∀?,
and Y
2n-1
=n. Both subsequences converge to
+∞

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- Fall '08
- Staff
- Limits, Limit of a sequence, Limit superior and limit inferior, lim sup, subsequence, Lim Sup Vn