Smallest Strictly Positive Rational Number does not Exist

Theorem
There exists no smallest element of the set of strictly positive rational numbers.

Proof
$x = \dfrac p q$ is the smallest strictly positive rational number.

By definition of strictly positive:
 * $0 < \dfrac p q$

Let us calculate the mediant of $0$ and $\dfrac p q$:


 * $\dfrac 0 1 < \dfrac {0 + p} {1 + q} < \dfrac p q$

The inequalities follow from Mediant is Between.

Thus $\dfrac p {1 + q}$ is a strictly positive rational number which is smaller than $\dfrac p q$.

Thus $\dfrac p q$ cannot be the smallest strictly positive rational number.

The result follows by Proof by Contradiction.