Dirichlet Function has no Period

Theorem
The Dirichlet functions are periodic by Dirichlet Function is Periodic.

However, they do not admit a period.

Proof
Let $D: \R \to \R$ be a Dirichlet function.

In proving that the Dirichlet Function is Periodic, it was shown that every non-zero rational number is a periodic element of $D$.

Therefore, the period of $D$ must be the smallest element of $\Q_{\gt 0}$.

But from Rational Numbers are not Well-Ordered under Conventional Ordering there cannot exist such an element.

Hence the result.