Real Numbers are Densely Ordered

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

Then $$\exists c \in \reals: a < c < b$$.

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

Proof
We can express $$a$$ and $$b$$ as $$a = \frac a 1, b = \frac b 1$$.

Thus from Mediant is Between:

$$\frac a 1 < \frac {a + b} {1 + 1} < \frac b 1$$

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