Squeeze Theorem
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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 f(x) is between g(x) and h(x) for all x in the neighborhood of S. If g and h approach some common limit L as x approaches S, then \lim_{x\to S}f(x)=L.
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.
For all 
, we must prove that there is some 
for which 
.
Now since, 
, there must exist 
such that,
and,
Now let 
. If 
, then
So 
. Now by the definition of a limit, we get 
.
