Real Numbers are Densely Ordered

Theorem
Let $a, b \in \R$ such that $a < b$.

Then:
 * $\exists c \in \R: a < c < b$

That is, the set of real numbers is close packed.

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