that $|f_\mu|_\lambda p^{c\cdot\mu}$ converges to $0$ as $|\mu|$ approaches $\infty$.

Note that $| \; |_\lambda$ is continuous, so $|f_\mu - f_\mu^{(i)}|_\lambda p^{c\cdot \mu} \leq 1/i$, for all $|\mu| \geq 0$ and all $i > 0$. We choose $\mu$, such that $|\mu|$ is sufficiently large, so that $|f_\mu^{(i)}|_\lambda p^{c\cdot \mu} < 1/i$. Since the norm is non-Archimedean, this shows that $|f_\mu|_\lambda p^{c\cdot \mu} \leq 1/i$. Thus, $|f_\mu|_\lambda p^{c\cdot \mu}$ converges to $0$ as $|\mu|$ approaches $\infty$. Hence, $f$ is an element of $F[X; Y, \lambda]_c$ and $\|f- f^{(i)}\|_{\lambda,c} = \max |f_\mu - f^{(i)}_\mu|_\lambda p^{c\cdot \mu} \leq  1/i$, therefore, $\lim_i f^{(i)} = f$.

A power series $f(X,Y) = \sum_{k=0}^\infty f_k(X,Y)X_n^k$ in $F[X; Y, \lambda]$ is called $X_n$-distinguished of degree $s$ in $F[X; Y, \lambda]$ if

1.  $f_s(X,Y)$ is a unit in $F[X_1,\cdots, X_{n-1}; Y, \lambda]$ and

2.  $|f_k(X, Y)| < 1$ for all $k > s$.

Equivalently, $f \mod (Y)$ is a unitary polynomial in $X_n$ of degree $s$.

A power series $f(X, Y) = \sum_{k=0}^\infty f_k(X,Y)X_n^k$ in $F[X; Y, \lambda]_c$ is called $X_n$-distinguished of degree $s$ in $F[X; Y, \lambda]_c$ if

1.  $f_s(X,Y)$ is a unit in $F[X_1,\cdots, X_{n-1}; Y, \lambda]_c$ and $\|f_s(X,Y)\|_{\lambda, (c_1, \cdots, c_{n-1})} = 1$,

2.  $\|f\|_{\lambda, c} 