Limit sup and limit inf.
Introduction
In order to make us understand the information more on approaches of a given real
sequence
a
n
n
1
, we give two definitions, thier names are upper limit and lower limit. It
is fundamental but important tools in analysis.
Definition of limit sup and limit inf
Definition
Given a real sequence
a
n
n
1
, we define
b
n
sup
a
m
:
m
n
and
c
n
inf
a
m
:
m
n
.
Example
1
1
n
n
1
0,2,0,2,...
, so we have
b
n
2 and
c
n
0 for all
n
.
Example
1
n
n
n
1
1,2,
3,4,...
, so we have
b
n
and
c
n
for all
n
.
Example
n
n
1
1,
2,
3,...
, so we have
b
n
n
and
c
n
for all
n
.
Proposition
Given a real sequence
a
n
n
1
, and thus define
b
n
and
c
n
as the same as
before.
1
b
n
, and
c
n
n
N
.
2
If there is a positive integer
p
such that
b
p
, then
b
n
n
N
.
If there is a positive integer
q
such that
c
q
, then
c
n
n
N
.
3
b
n
is decreasing and
c
n
is increasing.
By property 3, we can give definitions on the upper limit and the lower limit of a given
sequence as follows.
Definition
Given a real sequence
a
n
and let
b
n
and
c
n
as the same as before.
(1) If every
b
n
R
, then
inf
b
n
:
n
N
is called the upper limit of
a
n
, denoted by
lim
n
sup
a
n
.
That is,
lim
n
sup
a
n
inf
n
b
n
.
If every
b
n
, then we define
lim
n
sup
a
n
.
(2) If every
c
n
R
, then
sup
c
n
:
n
N
is called the lower limit of
a
n
, denoted by
lim
n
inf
a
n
.
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That is,
lim
n
inf
a
n
sup
n
c
n
.
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 Fall '09
 Frade
 lim, Supremum, Limit of a sequence, Limit superior and limit inferior, subsequence, n

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