Rational Numbers are Densely Ordered

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

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

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

Proof
From the definition of rational numbers, we can express $$a$$ and $$b$$ as $$a = \frac {p_1} {q_1}, b = \frac {p_2} {q_2}$$.

Thus from Mediant is Between:

$$\frac {p_1} {q_1} < \frac {p_1 + p_2} {q_1 + q_2} < \frac {p_2} {q_2}$$

As the rational numbers form a field, $$\frac {p_1 + p_2} {q_1 + q_2} \in \Q$$.

Hence $$c = \frac {p_1 + p_2} {q_1 + q_2}$$ is an element of $$\Q$$ between $$a$$ and $$b$$.