Squeeze Theorem
The Squeeze Theorem (also called the Sandwich Theorem or the Squeeze Play Theorem) is a relatively simple theorem that deals with calculus, specifically limits.
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[hide]Theorem
Suppose is between and for all in a neighborhood of the point . If and approach some common limit as approaches , then .
Proof
If is between and for all in the neighborhood of , then either or for all in this neighborhood. The two cases are the same up to renaming our functions, so assume without loss of generality that .
We must show that for all there is some for which implies .
Now since , there must exist such that
Now let . If then
So . Now by the definition of a limit we get as desired.
Applications and examples