I'm having troubles traying to understand the following theorem about the information of the Newton polygon of a power series over a nonarchimedean field and his zeros. The Theorem says:
Let K be a field complete with respect to a valuation (nonarchimedean absolute value) and let $f\in K[\![X]\!]$. Then to each finite side of the Newton polygon of $f$ there corresponds $l$ zeros $\alpha$ (counting multiplicities) of $f$ where $l$ is the length og the horizontal projection of the side. Moreover, if $\lambda$ is the slope of the side, then $\text{ord }\alpha=-\lambda$. Conversly if $\alpha$ is a root then $-\text{ord }\alpha$ is the slope of a (possibly infinite) side.
Here $\text{ord}$ is the additive valuation on $K$.
The proof relies on the factorizacion $f=PQ$, where $P$ is a polynomial having the zeros of $f$ corresponding to the given side, and $Q$ a power series having no one of the zeros of $P$. The proof constructs those elements by ''approximations'' given by Hensel lemma. In order to do that the author says:
It is sufficies to prove the assertion in the following special situation: the Newton polygon of $f$ contains a finite side with extreme vertices $(m,0)$ and $(m+n,0)$.
Then he explains how to do that: we start from a finite side of slope $\lambda$ whose horizontal projection has length $l$. We take a $\beta\in\Omega$ ($\Omega$ is the completion of the algebraic closure of $K$) with $\text{ord }\beta=-\lambda$. Then the series $f(\beta X)$ has a horizontal side of length $l$. Now the series $\delta f(\beta X)$, for suitable $\delta\neq 0$, has a side on the $x$-axis of length $l$.
My question is about the existence of such $\beta$, why does it exist? In general $\lambda$ may be any real number, and is not always possible to find such $\beta$. Even in the $p$-adics this is not the case: we always have rational slopes, but I think not necessarily there is and element in $\mathbb{C}_{p}$ with a given valuation. Also, the existence of such $\delta$ is not clear to me, since the vertices (the second coordinate of the vertex: $(i,\text{ord }a_{i})$) could have real coordinates.
Is this argument of the author correct?
Thanks