<s>
In	O
mathematics	O
,	O
the	O
limit	B-Algorithm
inferior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
superior	I-Algorithm
of	O
a	O
sequence	O
can	O
be	O
thought	O
of	O
as	O
limiting	O
(	O
that	O
is	O
,	O
eventual	O
and	O
extreme	O
)	O
bounds	O
on	O
the	O
sequence	O
.	O
</s>
<s>
They	O
can	O
be	O
thought	O
of	O
in	O
a	O
similar	O
fashion	O
for	O
a	O
function	O
(	O
see	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
)	O
.	O
</s>
<s>
In	O
general	O
,	O
when	O
there	O
are	O
multiple	O
objects	O
around	O
which	O
a	O
sequence	O
,	O
function	O
,	O
or	O
set	O
accumulates	O
,	O
the	O
inferior	O
and	O
superior	B-Algorithm
limits	I-Algorithm
extract	O
the	O
smallest	O
and	O
largest	O
of	O
them	O
;	O
the	O
type	O
of	O
object	O
and	O
the	O
measure	O
of	O
size	O
is	O
context-dependent	O
,	O
but	O
the	O
notion	O
of	O
extreme	O
limits	O
is	O
invariant	O
.	O
</s>
<s>
Limit	O
inferior	O
is	O
also	O
called	O
infimum	B-Algorithm
limit	I-Algorithm
,	O
limit	O
infimum	O
,	O
liminf	B-Algorithm
,	O
inferior	B-Algorithm
limit	I-Algorithm
,	O
lower	B-Algorithm
limit	I-Algorithm
,	O
or	O
inner	B-Algorithm
limit	I-Algorithm
;	O
limit	O
superior	O
is	O
also	O
known	O
as	O
supremum	B-Algorithm
limit	I-Algorithm
,	O
limit	O
supremum	O
,	O
limsup	B-Algorithm
,	O
superior	B-Algorithm
limit	I-Algorithm
,	O
upper	B-Algorithm
limit	I-Algorithm
,	O
or	O
outer	B-Algorithm
limit	I-Algorithm
.	O
</s>
<s>
If	O
the	O
terms	O
in	O
the	O
sequence	O
are	O
real	O
numbers	O
,	O
the	O
limit	B-Algorithm
superior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
inferior	I-Algorithm
always	O
exist	O
,	O
as	O
the	O
real	O
numbers	O
together	O
with	O
±∞	O
(	O
i.e.	O
</s>
<s>
Whenever	O
the	O
ordinary	O
limit	O
exists	O
,	O
the	O
limit	B-Algorithm
inferior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
superior	I-Algorithm
are	O
both	O
equal	O
to	O
it	O
;	O
therefore	O
,	O
each	O
can	O
be	O
considered	O
a	O
generalization	O
of	O
the	O
ordinary	O
limit	O
which	O
is	O
primarily	O
interesting	O
in	O
cases	O
where	O
the	O
limit	O
does	O
not	O
exist	O
.	O
</s>
<s>
The	O
limit	B-Algorithm
superior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
inferior	I-Algorithm
of	O
a	O
sequence	O
are	O
a	O
special	O
case	O
of	O
those	O
of	O
a	O
function	O
(	O
see	O
below	O
)	O
.	O
</s>
<s>
In	O
mathematical	O
analysis	O
,	O
limit	B-Algorithm
superior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
inferior	I-Algorithm
are	O
important	O
tools	O
for	O
studying	O
sequences	O
of	O
real	O
numbers	O
.	O
</s>
<s>
Assume	O
that	O
the	O
limit	B-Algorithm
superior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
inferior	I-Algorithm
are	O
real	O
numbers	O
(	O
so	O
,	O
not	O
infinite	O
)	O
.	O
</s>
<s>
The	O
relationship	O
of	O
limit	B-Algorithm
inferior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
superior	I-Algorithm
for	O
sequences	O
of	O
real	O
numbers	O
is	O
as	O
follows	O
:	O
</s>
<s>
The	O
liminf	B-Algorithm
and	O
limsup	B-Algorithm
of	O
a	O
sequence	O
are	O
respectively	O
the	O
smallest	O
and	O
greatest	O
cluster	O
points	O
.	O
</s>
<s>
In	O
the	O
particular	O
case	O
that	O
one	O
of	O
the	O
sequences	O
actually	O
converges	B-Algorithm
,	O
say	O
then	O
the	O
inequalities	O
above	O
become	O
equalities	O
(	O
with	O
or	O
being	O
replaced	O
by	O
)	O
.	O
</s>
<s>
As	O
in	O
the	O
case	O
for	O
sequences	O
,	O
the	O
limit	B-Algorithm
inferior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
superior	I-Algorithm
are	O
always	O
well-defined	O
if	O
we	O
allow	O
the	O
values	O
+∞	O
and	O
−∞	O
;	O
in	O
fact	O
,	O
if	O
both	O
agree	O
then	O
the	O
limit	O
exists	O
and	O
is	O
equal	O
to	O
their	O
common	O
value	O
(	O
again	O
possibly	O
including	O
the	O
infinities	O
)	O
.	O
</s>
<s>
The	O
difference	O
between	O
the	O
two	O
is	O
a	O
rough	O
measure	O
of	O
how	O
"	O
wildly	O
"	O
the	O
function	O
oscillates	O
,	O
and	O
in	O
observation	O
of	O
this	O
fact	O
,	O
it	O
is	O
called	O
the	O
oscillation	B-Algorithm
of	O
f	O
at	O
0	O
.	O
</s>
<s>
This	O
idea	O
of	O
oscillation	B-Algorithm
is	O
sufficient	O
to	O
,	O
for	O
example	O
,	O
characterize	O
Riemann-integrable	O
functions	O
as	O
continuous	O
except	O
on	O
a	O
set	O
of	O
measure	O
zero	O
.	O
</s>
<s>
Note	O
that	O
points	O
of	O
nonzero	O
oscillation	B-Algorithm
(	O
i.e.	O
,	O
points	O
at	O
which	O
f	O
is	O
"	O
badly	O
behaved	O
"	O
)	O
are	O
discontinuities	O
which	O
,	O
unless	O
they	O
make	O
up	O
a	O
set	O
of	O
zero	O
,	O
are	O
confined	O
to	O
a	O
negligible	O
set	O
.	O
</s>
<s>
There	O
is	O
a	O
notion	O
of	O
limsup	B-Algorithm
and	O
liminf	B-Algorithm
for	O
functions	O
defined	O
on	O
a	O
metric	O
space	O
whose	O
relationship	O
to	O
limits	O
of	O
real-valued	O
functions	O
mirrors	O
that	O
of	O
the	O
relation	O
between	O
the	O
limsup	B-Algorithm
,	O
liminf	B-Algorithm
,	O
and	O
the	O
limit	O
of	O
a	O
real	O
sequence	O
.	O
</s>
<s>
Hence	O
,	O
it	O
is	O
possible	O
(	O
and	O
sometimes	O
useful	O
)	O
to	O
consider	O
superior	O
and	O
inferior	B-Algorithm
limits	I-Algorithm
of	O
sequences	O
in	O
℘( X	O
)	O
(	O
i.e.	O
,	O
sequences	O
of	O
subsets	O
of	O
X	O
)	O
.	O
</s>
<s>
When	O
ordering	O
by	O
set	O
inclusion	O
,	O
the	O
supremum	B-Algorithm
limit	I-Algorithm
is	O
the	O
least	O
upper	O
bound	O
on	O
the	O
set	O
of	O
accumulation	O
points	O
because	O
it	O
contains	O
each	O
of	O
them	O
.	O
</s>
<s>
When	O
ordering	O
by	O
set	O
inclusion	O
,	O
the	O
infimum	B-Algorithm
limit	I-Algorithm
is	O
the	O
greatest	O
lower	O
bound	O
on	O
the	O
set	O
of	O
accumulation	O
points	O
because	O
it	O
is	O
contained	O
in	O
each	O
of	O
them	O
.	O
</s>
<s>
Because	O
ordering	O
is	O
by	O
set	O
inclusion	O
,	O
then	O
the	O
outer	B-Algorithm
limit	I-Algorithm
will	O
always	O
contain	O
the	O
inner	B-Algorithm
limit	I-Algorithm
(	O
i.e.	O
,	O
liminfXn	O
⊆	O
limsupXn	O
)	O
.	O
</s>
<s>
Hence	O
,	O
when	O
considering	O
the	O
convergence	O
of	O
a	O
sequence	O
of	O
sets	O
,	O
it	O
generally	O
suffices	O
to	O
consider	O
the	O
convergence	O
of	O
the	O
outer	B-Algorithm
limit	I-Algorithm
of	O
that	O
sequence	O
.	O
</s>
<s>
The	O
difference	O
between	O
the	O
two	O
definitions	O
involves	O
how	O
the	O
topology	B-Architecture
(	O
i.e.	O
,	O
how	O
to	O
quantify	O
separation	O
)	O
is	O
defined	O
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
second	O
definition	O
is	O
identical	O
to	O
the	O
first	O
when	O
the	O
discrete	O
metric	O
is	O
used	O
to	O
induce	O
the	O
topology	B-Architecture
on	O
X	O
.	O
</s>
<s>
which	O
is	O
also	O
called	O
the	O
inner	B-Algorithm
limit	I-Algorithm
,	O
consists	O
of	O
those	O
elements	O
which	O
are	O
limits	O
of	O
points	O
in	O
for	O
all	O
but	O
finitely	O
many	O
(	O
that	O
is	O
,	O
cofinitely	O
many	O
)	O
.	O
</s>
<s>
The	O
limit	O
exists	O
if	O
and	O
only	O
if	O
and	O
agree	O
,	O
in	O
which	O
case	O
The	O
outer	O
and	O
inner	B-Algorithm
limits	I-Algorithm
should	O
not	O
be	O
confused	O
with	O
the	O
set-theoretic	O
limits	O
superior	O
and	O
inferior	O
,	O
as	O
the	O
latter	O
sets	O
are	O
not	O
sensitive	O
to	O
the	O
topological	O
structure	O
of	O
the	O
space	O
.	O
</s>
<s>
That	O
is	O
,	O
this	O
case	O
specializes	O
the	O
general	O
definition	O
when	O
the	O
topology	B-Architecture
on	O
set	O
X	O
is	O
induced	O
from	O
the	O
discrete	O
metric	O
.	O
</s>
<s>
under	O
which	O
a	O
sequence	O
of	O
points	O
(	O
xk	O
)	O
converges	B-Algorithm
to	O
point	O
x	O
∈	O
X	O
if	O
and	O
only	O
if	O
xk	O
=	O
x	O
for	O
all	O
but	O
finitely	O
many	O
k	O
.	O
Therefore	O
,	O
if	O
the	O
limit	O
set	O
exists	O
it	O
contains	O
the	O
points	O
and	O
only	O
the	O
points	O
which	O
are	O
in	O
all	O
except	O
finitely	O
many	O
of	O
the	O
sets	O
of	O
the	O
sequence	O
.	O
</s>
<s>
In	O
this	O
context	O
,	O
the	O
inner	B-Algorithm
limit	I-Algorithm
,	O
liminfXn	O
,	O
is	O
the	O
largest	O
meeting	O
of	O
tails	O
of	O
the	O
sequence	O
,	O
and	O
the	O
outer	B-Algorithm
limit	I-Algorithm
,	O
limsupXn	O
,	O
is	O
the	O
smallest	O
joining	O
of	O
tails	O
of	O
the	O
sequence	O
.	O
</s>
<s>
They	O
have	O
been	O
broken	O
into	O
sections	O
with	O
respect	O
to	O
the	O
metric	O
used	O
to	O
induce	O
the	O
topology	B-Architecture
on	O
set	O
X	O
.	O
</s>
<s>
The	O
"	O
odd	O
"	O
and	O
"	O
even	O
"	O
elements	O
of	O
this	O
sequence	O
form	O
two	O
subsequences	O
,	O
( {	O
0}	O
,	O
{0},	O
{0},	O
...	O
)	O
and	O
( {	O
1}	O
,	O
{1},	O
{1},	O
...	O
)	O
,	O
which	O
have	O
limit	O
points	O
0	O
and	O
1	O
,	O
respectively	O
,	O
and	O
so	O
the	O
outer	O
or	O
superior	B-Algorithm
limit	I-Algorithm
is	O
the	O
set	O
 { 0 , 1 } 	O
of	O
these	O
two	O
points	O
.	O
</s>
<s>
However	O
,	O
there	O
are	O
no	O
limit	O
points	O
that	O
can	O
be	O
taken	O
from	O
the	O
(	O
Xn	O
)	O
sequence	O
as	O
a	O
whole	O
,	O
and	O
so	O
the	O
interior	O
or	O
inferior	B-Algorithm
limit	I-Algorithm
is	O
the	O
empty	O
set	O
{	O
}	O
.	O
</s>
<s>
That	O
is	O
,	O
the	O
four	O
elements	O
that	O
do	O
not	O
match	O
the	O
pattern	O
do	O
not	O
affect	O
the	O
liminf	B-Algorithm
and	O
limsup	B-Algorithm
because	O
there	O
are	O
only	O
finitely	O
many	O
of	O
them	O
.	O
</s>
<s>
So	O
long	O
as	O
the	O
tails	O
of	O
the	O
sequence	O
are	O
maintained	O
,	O
the	O
outer	O
and	O
inner	B-Algorithm
limits	I-Algorithm
will	O
be	O
unchanged	O
.	O
</s>
<s>
The	O
related	O
concepts	O
of	O
essential	O
inner	O
and	O
outer	B-Algorithm
limits	I-Algorithm
,	O
which	O
use	O
the	O
essential	O
supremum	O
and	O
essential	O
infimum	O
,	O
provide	O
an	O
important	O
modification	O
that	O
"	O
squashes	O
"	O
countably	O
many	O
(	O
rather	O
than	O
just	O
finitely	O
many	O
)	O
interstitial	O
additions	O
.	O
</s>
<s>
The	O
"	O
odd	O
"	O
and	O
"	O
even	O
"	O
elements	O
of	O
this	O
sequence	O
form	O
two	O
subsequences	O
,	O
( {	O
0}	O
,	O
{1/2},	O
{2/3},	O
{3/4},	O
...	O
)	O
and	O
( {	O
1}	O
,	O
{1/2},	O
{1/3},	O
{1/4},	O
...	O
)	O
,	O
which	O
have	O
limit	O
points	O
1	O
and	O
0	O
,	O
respectively	O
,	O
and	O
so	O
the	O
outer	O
or	O
superior	B-Algorithm
limit	I-Algorithm
is	O
the	O
set	O
 { 0 , 1 } 	O
of	O
these	O
two	O
points	O
.	O
</s>
<s>
However	O
,	O
there	O
are	O
no	O
limit	O
points	O
that	O
can	O
be	O
taken	O
from	O
the	O
(	O
Xn	O
)	O
sequence	O
as	O
a	O
whole	O
,	O
and	O
so	O
the	O
interior	O
or	O
inferior	B-Algorithm
limit	I-Algorithm
is	O
the	O
empty	O
set	O
{	O
}	O
.	O
</s>
<s>
The	O
Ω	O
limit	O
(	O
i.e.	O
,	O
limit	O
set	O
)	O
of	O
a	O
solution	O
to	O
a	O
dynamic	O
system	O
is	O
the	O
outer	B-Algorithm
limit	I-Algorithm
of	O
solution	O
trajectories	O
of	O
the	O
system	O
.	O
</s>
<s>
This	O
circle	O
,	O
which	O
is	O
the	O
limit	O
set	O
of	O
the	O
system	O
,	O
is	O
the	O
outer	B-Algorithm
limit	I-Algorithm
of	O
solution	O
trajectories	O
of	O
the	O
system	O
.	O
</s>
<s>
When	O
X	O
has	O
a	O
total	O
order	O
,	O
is	O
a	O
complete	O
lattice	O
and	O
has	O
the	O
order	B-Algorithm
topology	I-Algorithm
,	O
</s>
<s>
If	O
the	O
limit	B-Algorithm
inferior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
superior	I-Algorithm
agree	O
,	O
then	O
there	O
must	O
be	O
exactly	O
one	O
cluster	O
point	O
and	O
the	O
limit	O
of	O
the	O
filter	O
base	O
is	O
equal	O
to	O
this	O
unique	O
cluster	O
point	O
.	O
</s>
<s>
Therefore	O
,	O
these	O
definitions	O
give	O
the	O
limit	B-Algorithm
inferior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
superior	I-Algorithm
of	O
any	O
net	O
(	O
and	O
thus	O
any	O
sequence	O
)	O
as	O
well	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
limit	B-Algorithm
inferior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
superior	I-Algorithm
of	O
the	O
net	O
are	O
equal	O
to	O
the	O
limit	B-Algorithm
superior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
inferior	I-Algorithm
of	O
respectively	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
limit	B-Algorithm
inferior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
superior	I-Algorithm
of	O
the	O
sequence	O
are	O
equal	O
to	O
the	O
limit	B-Algorithm
superior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
inferior	I-Algorithm
of	O
respectively	O
.	O
</s>
