<s>
In	O
mathematics	O
,	O
a	O
sequence	O
of	O
functions	O
from	O
a	O
set	O
S	O
to	O
a	O
metric	O
space	O
M	O
is	O
said	O
to	O
be	O
uniformly	B-Algorithm
Cauchy	I-Algorithm
if	O
:	O
</s>
<s>
A	O
sequence	O
of	O
functions	O
 { fn } 	O
from	O
S	O
to	O
M	O
is	O
pointwise	O
Cauchy	O
if	O
,	O
for	O
each	O
x	O
S	O
,	O
the	O
sequence	O
{	O
fn(x )	O
}	O
is	O
a	O
Cauchy	O
sequence	O
in	O
M	O
.	O
This	O
is	O
a	O
weaker	O
condition	O
than	O
being	O
uniformly	B-Algorithm
Cauchy	I-Algorithm
.	O
</s>
<s>
In	O
general	O
a	O
sequence	O
can	O
be	O
pointwise	O
Cauchy	O
and	O
not	O
pointwise	O
convergent	O
,	O
or	O
it	O
can	O
be	O
uniformly	B-Algorithm
Cauchy	I-Algorithm
and	O
not	O
uniformly	B-Algorithm
convergent	I-Algorithm
.	O
</s>
<s>
Nevertheless	O
,	O
if	O
the	O
metric	O
space	O
M	O
is	O
complete	O
,	O
then	O
any	O
pointwise	O
Cauchy	O
sequence	O
converges	O
pointwise	O
to	O
a	O
function	O
from	O
S	O
to	O
M	O
.	O
Similarly	O
,	O
any	O
uniformly	B-Algorithm
Cauchy	I-Algorithm
sequence	I-Algorithm
will	O
tend	O
uniformly	B-Algorithm
to	O
such	O
a	O
function	O
.	O
</s>
<s>
Then	O
any	O
uniformly	B-Algorithm
Cauchy	I-Algorithm
sequence	I-Algorithm
of	O
continuous	O
functions	O
fn	O
:	O
S	O
M	O
tends	O
uniformly	B-Algorithm
to	O
a	O
unique	O
continuous	O
function	O
f	O
:	O
S	O
M	O
.	O
</s>
<s>
A	O
sequence	O
of	O
functions	O
from	O
a	O
set	O
S	O
to	O
a	O
uniform	O
space	O
U	O
is	O
said	O
to	O
be	O
uniformly	B-Algorithm
Cauchy	I-Algorithm
if	O
:	O
</s>
