Mediant is Between/Corollary 2/Proof 4

Proof
Let $\epsilon \in \R_{>0}$.

Then:
 * $\dfrac a b < \dfrac {c + \epsilon} d$

By Mediant is Between:
 * $\dfrac a b < \dfrac {a + c + \epsilon} {b + d} < \dfrac {c + \epsilon} d$

By Inequality of Sequences Preserved in Limit, letting $\epsilon \to 0$:
 * $\dfrac a b \le \dfrac {a + c} {b + d} \le \dfrac c d$

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

we have:
 * $\dfrac a b = \dfrac {a + c} {b + d} = \dfrac c d$