P-adic Norm and Absolute Value are Not Equivalent/Proof 1

Theorem
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$.

Let $\size{\,\cdot\,}$ be the absolute value on the rationals $\Q$.

Then $\norm {\,\cdot\,}_p$ and $\size{\,\cdot\,}$ are not equivalent norms.

That is, the topology induced by $\norm {\,\cdot\,}_p$ does not equal the topology induced by $\size{\,\cdot\,}$.

Proof
By definition of the $p$-adic norm:
 * $\norm{p}_p = \frac 1 p \lt 1$

By definition of the absolute value:
 * $\size{p} = p \gt 1$

By definition 4 of equivalent norms, $\norm {\,\cdot\,}_p$ and $\size{\,\cdot\,}$ are not equivalent norms.

By Equivalence of Definitions of Equivalent Division Ring Norms and definition 1 of equivalent norms then the topology induced by $\norm {\,\cdot\,}_p$ does not equal the topology induced by $\size{\,\cdot\,}$.