Mediant is Between/Corollary 1

Corollary to Mediant is Between

Let $a, b, c, d \in \Z$ be integers such that:

\(\text {(1)}: \quad\)

\(\ds b, d\)

\(>\)

\(\ds 0\)

\(\text {(2)}: \quad\)

\(\ds \dfrac a b\)

\(<\)

\(\ds \dfrac c d\)

Then the mediant of $\dfrac a b$ and $\dfrac c d$ is between $\dfrac a b$ and $\dfrac c d$:

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

Proof

By definition, $\dfrac a b$ and $\dfrac c d$ are rational numbers.

From Rational Numbers form Subfield of Real Numbers, $\dfrac a b, \dfrac c d \in \R$.

Hence from Mediant is Between:

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

$\blacksquare$