Squeeze Theorem/Functions/Proof 3

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Theorem

Let $a$ be a point on an open real interval $I$.

Also let $f$, $g$ and $h$ be real functions defined at all points of $I$ except for possibly at point $a$.

Suppose that:

$\forall x \ne a \in I: g \left({x}\right) \le f \left({x}\right) \le h \left({x}\right)$
$\displaystyle \lim_{x \mathop \to a} \ g \left({x}\right) = \lim_{x \mathop \to a} \ h \left({x}\right) = L$


Then:

$\displaystyle \lim_{x \mathop \to a} \ f \left({x}\right) = L$


Proof

By the definition of the limit of a real function, we have to prove that:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \left({\left|{x - a}\right| < \delta \implies \left|{f \left({x}\right) - L}\right| < \epsilon}\right)$


Let $\epsilon \in \R_{>0}$ be given.

We have:

$\displaystyle \lim_{x \mathop \to a} \ g \left({x}\right) = \lim_{x \mathop \to a} \ h \left({x}\right)$

Hence by Sum Rule for Limits of Functions:

$\displaystyle \lim_{x \mathop \to a} \ h \left({x}\right) - g \left({x}\right) = 0$


By the definition of the limit of a real function:

$(1): \quad \forall \epsilon' \in \R_{>0}: \exists \delta \in \R_{>0}: \left({\left|{x - a}\right| < \delta \implies \left|{h \left({x}\right) - L}\right| < \epsilon'}\right)$
$(2): \quad \forall \epsilon' \in \R_{>0}: \exists \delta \in \R_{>0}: \left({\left|{x - a}\right| < \delta \implies \left|{g \left({x}\right) - L}\right| < \epsilon'}\right)$
$(3): \quad \forall \epsilon' \in \R_{>0}: \exists \delta \in \R_{>0}: \left({\left|{x - a}\right| < \delta \implies \left|{h \left({x}\right) - g \left({x}\right)}\right| < \epsilon'}\right)$


Take $\epsilon' = \dfrac {\epsilon} 3$ in $(1)$, $(2)$, $(3)$.

Then there exists $\delta_1, \delta_2, \delta_3$ that satisfies $(1)$, $(2)$, $(3)$ with $\epsilon' = \dfrac \epsilon 3$.

Take $\delta = \min\{\delta_1, \delta_2, \delta_3\}$.

Then:

\(\displaystyle \left\vert{x - a}\right\vert\) \(<\) \(\displaystyle \delta\)
\(\displaystyle \implies \ \ \) \(\displaystyle \left\vert{h \left({x}\right) - L}\right\vert\) \(<\) \(\displaystyle \frac {\epsilon} 3\)
\(\, \displaystyle \land \, \) \(\displaystyle \left\vert{g \left({x}\right) - L}\right\vert\) \(<\) \(\displaystyle \frac {\epsilon} 3\)
\(\, \displaystyle \land \, \) \(\displaystyle \left\vert{h \left({x}\right) - g \left({x}\right)}\right\vert\) \(<\) \(\displaystyle \frac {\epsilon} 3\)

So, if $\left\vert{x - a}\right\vert<\delta$:

\(\displaystyle \left\vert{f \left({x}\right) - L}\right\vert\) \(=\) \(\displaystyle \left\vert{f \left({x}\right) - g \left({x}\right) + h \left({x}\right) - L + g \left({x}\right) - h \left({x}\right)}\right\vert\)
\(\displaystyle \) \(\le\) \(\displaystyle \left\vert{f \left({x}\right) - g \left({x}\right)}\right\vert + \left\vert{h \left({x}\right) - L}\right\vert + \left\vert{h \left({x}\right) - g \left({x}\right)}\right\vert\)
\(\displaystyle \) \(\le\) \(\displaystyle \left\vert{h \left({x}\right) - g \left({x}\right)}\right\vert + \left\vert{h \left({x}\right) - L}\right\vert + \left\vert{h \left({x}\right) - g \left({x}\right)}\right\vert\)
\(\displaystyle \) \(\le\) \(\displaystyle \frac {\epsilon} 3 + \frac {\epsilon} 3 + \frac {\epsilon} 3\)
\(\displaystyle \) \(=\) \(\displaystyle \epsilon\)

$\blacksquare$