Squeeze Theorem/Functions/Proof 3

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Theorem

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

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

Suppose that:

$\forall x \ne a \in I: \map g x \le \map f x \le \map h x$
$\ds \lim_{x \mathop \to a} \map g x = \lim_{x \mathop \to a} \map h x = L$


Then:

$\ds \lim_{x \mathop \to a} \ \map f x = L$


Proof

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

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \paren {\size {x - a} < \delta \implies \size {\map f x - L} < \epsilon}$


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

We have:

$\ds \lim_{x \mathop \to a} \map g x = \lim_{x \mathop \to a} \map h x$

Hence by Sum Rule for Limits of Real Functions:

$\ds \lim_{x \mathop \to a} \paren {\map h x - \map g x} = 0$


By the definition of the limit of a real function:

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


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

Then:

\(\ds \size {x - a}\) \(<\) \(\ds \delta\)
\(\ds \leadsto \ \ \) \(\ds \size {\map h x - L}\) \(<\) \(\ds \frac {\epsilon} 3\)
\(\, \ds \land \, \) \(\ds \size {\map g x - L}\) \(<\) \(\ds \frac {\epsilon} 3\)
\(\, \ds \land \, \) \(\ds \size {\map h x - \map g x}\) \(<\) \(\ds \frac {\epsilon} 3\)

So, if $\size {x - a} < \delta$:

\(\ds \size {\map f x - L}\) \(=\) \(\ds \size {\map f x - \map g x + \map h x - L + \map g x - \map h x}\)
\(\ds \) \(\le\) \(\ds \size {\map f x - \map g x} + \size {\map h x - L} + \size {\map h x - \map g x}\)
\(\ds \) \(\le\) \(\ds \size {\map h x - \map g x} + \size {\map h x - L} + \size {\map h x - \map g x}\)
\(\ds \) \(\le\) \(\ds \frac {\epsilon} 3 + \frac {\epsilon} 3 + \frac {\epsilon} 3\)
\(\ds \) \(=\) \(\ds \epsilon\)

$\blacksquare$



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