Norms Equivalent to Absolute Value on Rational Numbers/Necessary Condition

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

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 $\Q$.

Let $\norm {\,\cdot\,}$ be a norm on $\Q$.

Then:
 * $\alpha \le 1$

Proof
The contrapositive is proved.

Let $\alpha > 1$.

The is not satisfied:

By Rule of Transposition the result follows.