(See also Spivak Chapter 8 - Problem 18) Let {xk} be a
bounded sequence. We define the limit superior (limsup) and limit
inferior (liminf)
of the
sequence to be
limsup
k → ∞
xk
=
lim
k→∞
⎛ ⎝
sup
n ≥ k
xn
⎞ ⎠
,
liminf
k → ∞
xk
=
lim
k→∞
⎛ ⎝
inf
n ≥ k
xn
⎞ ⎠
.
•
(P13-BISHL) shows that both limsupk → ∞xk and liminfk → ∞ xk exist.
•
Show that
limsup
k → ∞
xk
= A
if and only if for every ϵ > 0,
⎧ ⎪ ⎨
⎪ ⎩
xk > A+ ϵ
foratmostfinitelymanyk,
xk > A − ϵ
forinfinitelymanyk.
•
Show that if {xk} and {yk} are bounded
sequences, then
limsup
k → ∞
(xk + yk) ≤
limsup
k → ∞
xk +
limsup
k → ∞
yk.
•
Give an example of bounded sequences {xk} and {yk}
such that
limsup
k → ∞
(xk + yk) <
limsup
k → ∞
xk +
limsup
k → ∞
yk.
•
Show that if {xk} and {yk} are bounded
sequences, then
liminf
k → ∞
xk +
limsup
k → ∞
yk ≤
limsup
k → ∞
(xk + yk).
•
The general result is that for two bounded sequences {xk} and
{yk},
liminf
k → ∞
xk +
liminf
k → ∞
yk
≤
liminf
k → ∞
(xk + yk)
≤
liminf
k → ∞
xk +
limsup
k → ∞
yk
≤
limsup
k → ∞
(xk + yk)
≤
limsup
k → ∞
xk +
limsup
k → ∞
yk.
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version 4.04. On 22 Aug 2014, 01:37.