Archimedean Valuation

A Valuation for which $\vert x\vert \leq 1$ Implies $\vert 1+x\vert \leq C$ for the constant $C= 1$ (independent of $x$). Such a Valuation does not satisfy the strong Triangle Inequality

$$\vert x+y\vert \leq \max (\vert x\vert, \vert y\vert).$$