Norms Equivalent to Absolute Value on Rational Numbers/Necessary Condition

Theorem
Let $\alpha \in \R_{\gt 0}$.

Let $\norm{\,\cdot\,}:\Q \to \R$ be the mapping defined by:
 * $\forall x \in \Q: \norm{x} = \size {x}^\alpha$

where $\size {x}$ is the absolute value of $x$ in $\Q$.

Then:
 * $\norm{\,\cdot\,}$ is a norm on $\Q \implies \,\,\alpha \le 1$

Proof
The contrapositive is proved.

Let $\alpha \gt 1$.

The norm axiom (N3) (Triangle Inequality) is not satisfied:

By Rule of Transposition the result follows.