On page 73 of 'Lie algebras and Lie groups', Serre proves the inverse function theorem for complete fields. I would like to have some clarification about the following point.
Let $K$ be a complete ultrametric field. Let $f = (f_1,\dots,f_n)$ be an $n$-uple of formal power series in $n$ variables (i.e. $f_i \in K[[X_1, \dots,X_n]]$). Suppose that $f$ is convergent and that $D_0 f$ is invertible.
Serre proves that there exists $g=(g_1,\dots,g_n)$ a formal power series convergent satisfying : $f \circ g(\underline{T}) = \underline{T} = (T_1,\dots,T_n)$ as formal power series (and also satisfying $g \circ f(\underline{T}) = \underline{T} = (T_1,\dots,T_n)$ if I am not mistaken). Serre ends the proof here.
Why does this prove the theorem ?
To conclude, we have to prove that there exist two opens $U,V$ such that $f$ maps U into $V$ and $g$ maps $V$ into $U$. But I can't see why these opens exist. It seems to me that we have to prove $f$ is an open map.