Real Numbers are Densely Ordered/Proof 1

Proof
We can express $a$ and $b$ as:
 * $a = \dfrac a 1, b = \dfrac b 1$

Thus from Mediant is Between:
 * $\dfrac a 1 < \dfrac {a + b} {1 + 1} < \dfrac b 1$

Hence $c = \dfrac {a + b} 2$ is an element of $\R$ between $a$ and $b$.