Mediant is Between/Corollary 2/Proof 1

Proof
Let $p, q, r, s \in \R$ such that $q > 0, s > 0$.

Then from Mediant is Between:


 * $\dfrac p q < \dfrac {p + r} {q + s} < \dfrac r s$

In order to present this in the form required by the Squeeze Theorem for Functions, we weaken the ordering:


 * $(1): \quad \dfrac p q \le \dfrac {p + r} {q + s} \le \dfrac r s$

Let $f$, $g$ and $h$ be the real functions defined as:

From $(1)$:
 * $\forall x \in \R: \map f x \le \map g x \le \map h x$

But :
 * $\dfrac a b = \dfrac c d$

That is:
 * $\forall x \in \R: \map f x = \map h x$

Hence from the Squeeze Theorem for Functions:


 * $\forall x \in \R: \map f x = \map g x = \map h x$

That is:
 * $\dfrac a b = \dfrac c d \implies \dfrac a b = \dfrac {a + c} {b + d} = \dfrac c d$