Definition:Archimedean Property/Norm

Definition
Let $\struct {S, \circ}$ be a closed algebraic structure.

Let $\cdot: \Z_{>0} \times S \to S$ be the operation defined as:
 * $m \cdot a = \begin{cases}

a & : m = 1 \\ a \circ \paren {\paren {m - 1} \cdot a} & : m > 1 \end {cases}$

Let $n: S \to \R$ be a norm on $S$.

Then $n$ satisfies the Archimedean property on $S$ :
 * $\forall a, b \in S: n \paren a < n \paren b \implies \exists m \in \N: n \paren {m \cdot a} > n \paren b$

Using the more common symbology for a norm:
 * $\forall a, b \in S: \norm a < \norm b \implies \exists m \in \Z_{>0}: \norm {m \cdot a} > \norm b$