<s>
In	O
mathematics	O
,	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
sequence	I-Algorithm
is	O
the	O
value	O
that	O
the	O
terms	O
of	O
a	O
sequence	O
"	O
tend	O
to	O
"	O
,	O
and	O
is	O
often	O
denoted	O
using	O
the	O
symbol	O
(	O
e.g.	O
,	O
)	O
.	O
</s>
<s>
The	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
sequence	I-Algorithm
is	O
said	O
to	O
be	O
the	O
fundamental	O
notion	O
on	O
which	O
the	O
whole	O
of	O
mathematical	O
analysis	O
ultimately	O
rests	O
.	O
</s>
<s>
Leucippus	O
,	O
Democritus	O
,	O
Antiphon	O
,	O
Eudoxus	O
,	O
and	O
Archimedes	B-Device
developed	O
the	O
method	O
of	O
exhaustion	O
,	O
which	O
uses	O
an	O
infinite	O
sequence	O
of	O
approximations	O
to	O
determine	O
an	O
area	O
or	O
a	O
volume	O
.	O
</s>
<s>
Archimedes	B-Device
succeeded	O
in	O
summing	O
what	O
is	O
now	O
called	O
a	O
geometric	O
series	O
.	O
</s>
<s>
Finding	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
sequence	I-Algorithm
is	O
not	O
always	O
obvious	O
.	O
</s>
<s>
The	O
squeeze	B-Algorithm
theorem	I-Algorithm
is	O
often	O
useful	O
in	O
the	O
establishment	O
of	O
such	O
limits	O
.	O
</s>
<s>
When	O
it	O
exists	O
,	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
sequence	I-Algorithm
is	O
unique	O
.	O
</s>
<s>
(	O
Squeeze	B-Algorithm
theorem	I-Algorithm
)	O
If	O
for	O
all	O
greater	O
than	O
some	O
,	O
and	O
,	O
then	O
.	O
</s>
<s>
(	O
Monotone	O
convergence	B-Algorithm
theorem	O
)	O
If	O
is	O
bounded	O
and	O
monotonic	O
for	O
all	O
greater	O
than	O
some	O
,	O
then	O
it	O
is	O
convergent	O
.	O
</s>
<s>
However	O
,	O
a	O
divergent	B-Algorithm
sequence	I-Algorithm
need	O
not	O
tend	O
to	O
plus	O
or	O
minus	O
infinity	O
,	O
and	O
the	O
sequence	O
provides	O
one	O
such	O
example	O
.	O
</s>
<s>
When	O
it	O
exists	O
,	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
sequence	I-Algorithm
is	O
unique	O
,	O
as	O
distinct	O
points	O
are	O
separated	O
by	O
some	O
positive	O
distance	O
,	O
so	O
for	O
less	O
than	O
half	O
this	O
distance	O
,	O
sequence	O
terms	O
cannot	O
be	O
within	O
a	O
distance	O
of	O
both	O
points	O
.	O
</s>
<s>
One	O
particularly	O
important	O
result	O
in	O
real	O
analysis	O
is	O
the	O
Cauchy	O
criterion	O
for	O
convergence	B-Algorithm
of	O
sequences	O
:	O
a	O
sequence	O
of	O
real	O
numbers	O
is	O
convergent	O
if	O
and	O
only	O
if	O
it	O
is	O
a	O
Cauchy	O
sequence	O
.	O
</s>
<s>
A	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
sequence	I-Algorithm
of	O
points	O
in	O
a	O
topological	O
space	O
is	O
a	O
special	O
case	O
of	O
a	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
:	O
the	O
domain	B-Algorithm
is	O
in	O
the	O
space	O
,	O
with	O
the	O
induced	O
topology	O
of	O
the	O
affinely	O
extended	O
real	O
number	O
system	O
,	O
the	O
range	B-Algorithm
is	O
,	O
and	O
the	O
function	O
argument	O
tends	O
to	O
,	O
which	O
in	O
this	O
space	O
is	O
a	O
limit	O
point	O
of	O
.	O
</s>
<s>
Note	O
that	O
the	O
double	O
limit	O
is	O
different	O
from	O
taking	O
limit	O
in	O
n	O
first	O
,	O
and	O
then	O
in	O
m	O
.	O
The	O
latter	O
is	O
known	O
as	O
iterated	B-Algorithm
limit	I-Algorithm
.	O
</s>
<s>
Given	O
that	O
both	O
the	O
double	O
limit	O
and	O
the	O
iterated	B-Algorithm
limit	I-Algorithm
exists	O
,	O
they	O
have	O
the	O
same	O
value	O
.	O
</s>
<s>
However	O
,	O
a	O
divergent	B-Algorithm
sequence	I-Algorithm
need	O
not	O
tend	O
to	O
plus	O
or	O
minus	O
infinity	O
,	O
and	O
the	O
sequence	O
provides	O
one	O
such	O
example	O
.	O
</s>
<s>
When	O
such	O
a	O
limit	O
exists	O
,	O
we	O
say	O
the	O
sequence	O
converges	B-Algorithm
pointwise	I-Algorithm
to	O
.	O
</s>
<s>
Hence	O
,	O
one	O
can	O
easily	O
see	O
that	O
uniform	B-Algorithm
convergence	I-Algorithm
is	O
a	O
stronger	O
property	O
than	O
pointwise	B-Algorithm
convergence	I-Algorithm
:	O
the	O
existence	O
of	O
uniform	B-Algorithm
limit	I-Algorithm
implies	O
the	O
existence	O
and	O
equality	O
of	O
pointwise	O
limit	O
:	O
</s>
<s>
When	O
such	O
a	O
limit	O
exists	O
,	O
we	O
say	O
the	O
sequence	O
converges	B-Algorithm
uniformly	I-Algorithm
to	O
.	O
</s>
<s>
This	O
limit	O
is	O
known	O
as	O
iterated	B-Algorithm
limit	I-Algorithm
of	O
the	O
double	O
sequence	O
.	O
</s>
