MthT 430 Projects Chap 8f Note - limsup and liminf
MthT 430 Projects Chap 8f Note - limsup and liminf
Really Understanding sup and inf
•
(See also Spivak Chapter 8 - Problem 18) Let {xk} be a bounded sequence. We define the limit superior and limit inferior 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.
In http://www.math.uic.edu/~jlewis/mtht430/2007project.pdf, we made the definition



Definition. If f is a bounded function on [0,1], we define
sup
f
=
sup
x ∈ [0,1] 
 f(x),
inf
f
=
inf
x ∈ [0,1] 
 f(x).
We could just replace the domain of the function f, [0,1], by a set of natural numbers Zk = {k, k+1,…}, and use the usual conventions for sequences to make the definition:

Definition. If {xk} is a bounded sequence of real numbers, we define
x
 
k 
=
sup
n ≥ k 
 xn,
xk
=
inf
n ≥ k 
 xn.
Note that {―xk} is a nonincreasing sequence which is bounded below so that

limsup
k → ∞ 
 xk =
lim
k→∞ 
x
 
k 
 =
inf
k 
x
 
k 
exists. Similarly {xk} is a nondecreasing sequence which is bounded above so that

liminf
k → ∞ 
 xk =
lim
k→∞ 
 xk =
sup
k 
 xk
exists.
N.B. For every k,
xk
≤
x
 
k 
 .
It follows that


liminf
k→∞ 
 xk
≤
limsup
k→∞ 
 xk
Necessary and Sufficient Condition (NASC) for Existence of a Limit of a Sequence
•
Show that

limsup
k → ∞ 
 xk
= A
if and only if for every ϵ > 0,
⎧
⎪
⎨
⎪
⎩
xk > A + ϵ
for at most finitely many k,
xk > A − ϵ
for infinitely many k.
•
Show that

liminf
k → ∞ 
 xk
= A
if and only if for every ϵ > 0,
⎧
⎪
⎨
⎪
⎩
xk < A − ϵ
for at most finitely many k,
xk < A + ϵ
for infinitely many k.
•
Prove:

Theorem. Let {xk} be a bounded sequence. Then

lim
k→∞ 
 xk exists
if and only if

liminf
k→∞ 
 xk
=
limsup
k→∞ 
 xk.
Proof. If limk → ∞xk = L, show that liminfk → ∞xk = limk →∞xk., …. For the converse, assume liminfk→∞xk = A = limsupk→∞xk. Given ϵ > 0, except for at most finitely many k, A − ϵ < xk < A + ϵ.
Inequalities with limsup and liminf
See http://www.math.uic.edu/~jlewis/mtht430/2007project.pdf
N.B. Let {xk} and {yk} be bounded sequences. Following the proof of Problem 10 in 2007project.pdf, for each k have

inf
n ≥ k 
 xn +
inf
n ≥ k 
 yn
≤
inf
n ≥ k 
 (xn + yn)
≤
inf
n ≥ k 
 xn +
sup
n ≥ k 
 yn
≤
sup
n ≥ k 
 (xn + yn)
≤
sup
n ≥ k 
 xn +
sup
n ≥ k 
 yn.
•
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.
•
The Monster Counterexample to Equality:
{xk}
= {2,2,0,0,2,2,0,0,…},
{yk}
= {0,1,1,2,0,1,1,2,…},
{xk + yk}
= {2,3,1,2,2,3,1,2,…}


File translated from TEX by TTH, version 4.04.
On 22 Aug 2014, 01:32.