Mediant is Between

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Theorem

Let $r, s \in \Q$, i.e. let $r, s$ be rational numbers.

Then the mediant of $r$ and $s$ is between $r$ and $s$.

Let $a, b, c, d$ be any real numbers such that $b > 0, d > 0$.

Let $r = \dfrac a b < \dfrac c d = s$.

Then:

$r < \dfrac {a + c} {b + d} < s$

The same proof can apply to both.

Let $r, s \in \R$ be such that $r < s$ and $r = \dfrac a b, s = \dfrac c d$, where $a, b, c, d$ are real numbers such that $b > 0, d > 0$.

Because $b, d > 0$, it follows that $b d > 0$ from Real Number Ordering is Compatible with Multiplication. Thus we have:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \frac a b$	$<$	$\displaystyle \frac c d$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle \frac a b \left({b d}\right)$	$<$	$\displaystyle \frac c d \left({b d}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Real Number Ordering is Compatible with Multiplication
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle a d$	$<$	$\displaystyle b c$	$\displaystyle $	$\displaystyle $	$\displaystyle $

Then:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle a \left({b + d}\right)$	$=$	$\displaystyle a b + a d$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$<$	$\displaystyle a b + b c$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Real Number Ordering is Compatible with Addition, as $ad < bc$ from above
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left({a + c}\right) b$	$\displaystyle $	$\displaystyle $	$\displaystyle $

From Inverse of Positive Real is Positive, $\left({a + c}\right)^{-1} > 0$ and $b^{-1} > 0$.

It follows from Ordering is Compatible with Multiplication that $\dfrac a b < \dfrac {a + c} {b + d}$.

The other half is proved similarly.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \frac a b\)	\(<\)	\(\displaystyle \frac c d\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \frac a b \left({b d}\right)\)	\(<\)	\(\displaystyle \frac c d \left({b d}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Real Number Ordering is Compatible with Multiplication
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle a d\)	\(<\)	\(\displaystyle b c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle a \left({b + d}\right)\)	\(=\)	\(\displaystyle a b + a d\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(<\)	\(\displaystyle a b + b c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Real Number Ordering is Compatible with Addition, as $ad < bc$ from above
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({a + c}\right) b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)