Limit of Function in Interval

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Theorem

Let $f$ be a real function which is defined on the open interval $\openint a b$.

Let $\xi \in \openint a b$

Suppose that, $\forall x \in \openint a b$, either:

$\xi \le \map f x \le x$

or:

$x \le \map f x \le \xi$


Then $\map f x \to \xi$ as $x \to \xi$.


Proof

Note that $\size {\map f x - \xi} \le \size {\xi - x}$.

From Limit of Absolute Value‎ we have that $\size {x - \xi} \to 0$ as $x \to \xi$.

The result follows from the Squeeze Theorem.

$\blacksquare$


Sources