Rational Numbers are Close Packed

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 $a = \dfrac {p_1} {q_1}, b = \dfrac {p_2} {q_2}$.

Thus from Mediant is Between:

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

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

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

$\blacksquare$