Squeeze Theorem
This is an AoPSWiki Word of the Week for May 4-11 |
The Squeeze Theorem (also called the Sandwich Theorem or the Squeeze Play Theorem) is a relatively simple theorem that deals with calculus, specifically limits.
Theorem
Suppose is between
and
for all
in the neighborhood of
. If
and
approach some common limit L as
approaches
, then
.
Proof
If is between
and
for all
in the neighborhood of
, then either
or
for all
in the neighborhood of
. Since the second case is basically the first case, we just need to prove the first case.
If increases to
, then
goes to either
or
, where
. If
decreases to
, then
goes to either
or
, where
. Since
can't go to
or
, then
must go to
. Therefore,
.