The image of a valuation is dense in $\mathbb{R}$

2018-03-12 20:21:43

I'm reading Dwork's Boook An introduction to G-Functions and I'm stuck in some part of a proposition.

We say that $|-|$ is a valuation in the field $K$ (With values in $\mathbb{R}_{\geq 0}$) if

1) $|x|=0$ if and only if $x=0$

2) For all $x,y\in K$, $|xy|=|x||y|$, and

3) $|x+y|\leq\max\{|x|,|y|\}$.

We define the valuation group of $|-|$ by $G=\text{Im}|-|$. In the proof of the proposition which I'm stucked the key is to show that $G$ is dense in $\mathbb{R}$, but I cannot see why.

He says: '' Fix $\alpha\in K, \alpha\neq 0$ such that $|\alpha|\leq\varepsilon$'', where $\varepsilon$ is a fixed number greather than zero.

Can anyone give me any hint?

Thanks