Smallest Strictly Positive Rational Number does not Exist
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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 $1$