Let $K$ be a field of characteristic zero complete with respect to a non archimedian absolute value with a residue field of characteristic $p>0$. Let $\mathcal{H}^\dagger=\cup_{\varepsilon >0} \mathcal{A}(]1-\varepsilon, \infty])$. Where $\mathcal{A}(]1-\varepsilon,\infty])$ is the set of analytic function on $]1-\varepsilon,\infty]$. I would like to prove that if we take $m \in 1+\frac{1}{x}\mathcal{H}^\dagger$ then we can write $m$ in the form $$\prod\limits_{i\geq 1} (1+\frac{\mu_i}{x^i}).$$
Thanks for the help !