Smallest Strictly Positive Rational Number does not Exist

Theorem

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

Proof

Aiming for a contradiction, suppose $x = \dfrac p q$ 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.

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: Exercise $1$