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