<s>
In	O
calculus	O
,	O
the	O
squeeze	B-Algorithm
theorem	I-Algorithm
(	O
also	O
known	O
as	O
the	O
sandwich	B-Algorithm
theorem	I-Algorithm
,	O
among	O
other	O
names	O
)	O
is	O
a	O
theorem	O
regarding	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
that	O
is	O
trapped	O
between	O
two	O
other	O
functions	O
.	O
</s>
<s>
The	O
squeeze	B-Algorithm
theorem	I-Algorithm
is	O
used	O
in	O
calculus	O
and	O
mathematical	O
analysis	O
,	O
typically	O
to	O
confirm	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
via	O
comparison	O
with	O
two	O
other	O
functions	O
whose	O
limits	O
are	O
known	O
.	O
</s>
<s>
It	O
was	O
first	O
used	O
geometrically	O
by	O
the	O
mathematicians	O
Archimedes	B-Device
and	O
Eudoxus	O
in	O
an	O
effort	O
to	O
compute	O
,	O
and	O
was	O
formulated	O
in	O
modern	O
terms	O
by	O
Carl	O
Friedrich	O
Gauss	O
.	O
</s>
<s>
French	O
,	O
German	O
,	O
Italian	O
,	O
Hungarian	O
and	O
Russian	O
)	O
,	O
the	O
squeeze	B-Algorithm
theorem	I-Algorithm
is	O
also	O
known	O
as	O
the	O
two	O
officers	O
(	O
and	O
a	O
drunk	O
)	O
theorem	O
,	O
or	O
some	O
variation	O
thereof	O
.	O
</s>
<s>
The	O
squeeze	B-Algorithm
theorem	I-Algorithm
is	O
formally	O
stated	O
as	O
follows	O
.	O
</s>
<s>
According	O
to	O
the	O
above	O
hypotheses	O
we	O
have	O
,	O
taking	O
the	O
limit	B-Algorithm
inferior	O
and	O
superior	O
:	O
</s>
<s>
A	O
direct	O
proof	O
,	O
using	O
the	O
-definition	O
of	O
limit	B-Algorithm
,	O
would	O
be	O
to	O
prove	O
that	O
for	O
all	O
real	O
there	O
exists	O
a	O
real	O
such	O
that	O
for	O
all	O
with	O
,	O
we	O
have	O
.	O
</s>
<s>
The	O
proof	O
for	O
sequences	O
is	O
very	O
similar	O
,	O
using	O
the	O
-definition	O
of	O
the	O
limit	B-Algorithm
of	O
a	O
sequence	O
.	O
</s>
<s>
Since	O
,	O
by	O
the	O
squeeze	B-Algorithm
theorem	I-Algorithm
,	O
must	O
also	O
be	O
0	O
.	O
</s>
<s>
The	O
squeeze	B-Algorithm
theorem	I-Algorithm
can	O
still	O
be	O
used	O
in	O
multivariable	O
calculus	O
but	O
the	O
lower	O
(	O
and	O
upper	O
functions	O
)	O
must	O
be	O
below	O
(	O
and	O
above	O
)	O
the	O
target	O
function	O
not	O
just	O
along	O
a	O
path	O
but	O
around	O
the	O
entire	O
neighborhood	O
of	O
the	O
point	O
of	O
interest	O
and	O
it	O
only	O
works	O
if	O
the	O
function	O
really	O
does	O
have	O
a	O
limit	B-Algorithm
there	O
.	O
</s>
<s>
It	O
can	O
,	O
therefore	O
,	O
be	O
used	O
to	O
prove	O
that	O
a	O
function	O
has	O
a	O
limit	B-Algorithm
at	O
a	O
point	O
,	O
but	O
it	O
can	O
never	O
be	O
used	O
to	O
prove	O
that	O
a	O
function	O
does	O
not	O
have	O
a	O
limit	B-Algorithm
at	O
a	O
point	O
.	O
</s>
<s>
therefore	O
,	O
by	O
the	O
squeeze	B-Algorithm
theorem	I-Algorithm
,	O
</s>
