= \frac{1}{p^{{\text{ord}}_Y(f-f^{(i)})}}$ converges to $0$ as $i$ approaches $\infty$, and so ${\text{ord}}_Y(f - f^{(i)})$ must approach $\infty$. This can happen if, and only if, the corresponding sequence, $\lambda({\text{ord}}_Y(f - f^{(i)}))$, approaches $\infty$, as desired.

A norm which only satisfies $|ab| \leq |a||b|$, instead of strict equality is sometimes called a pseudo-norm; we disregard the distinction in this paper.

We can write any element $f$ in $F[X; Y, \lambda]$ in the following form: $$\begin{aligned}
 f(X,Y) &=& \sum_{\mu} f_\mu(Y)X^\mu\end{aligned}$$ where $\mu = (\mu_1, \cdots, \mu_n)$ is a tuple of positive integers, and $X^\mu = X_1^{\mu_1}\cdots X_n^{\mu_n}$. This form and the above norms allow us to formulate two equivalent definitions for $F[X; Y, \lambda]$: $$\begin{aligned}
 F[X; Y, \lambda] &=& \{f = \sum_{\mu}^\infty f_{\mu}( Y)X^\mu : f_\mu  \in F[[Y]], |f_\mu| p^{\lambda^{-1}(C_f|\mu|)} \stackrel{|\mu| \rightarrow \infty}{\rightarrow} 0\} \\
&=& \{f = \sum_{\mu}^\infty f_{\mu}( Y)X^\mu : f_\mu  \in F[[Y]], |f_\mu|_\lambda p^{C_f|\mu|} \stackrel{|\mu| \rightarrow \infty}{\rightarrow} 0\}\end{aligned}$$ where $|\mu| = \mu_1 + \cdots + \mu_n$.

For all $c$ in ${\mathbf{R}}^{n}, c_i > 0$, define $$\begin{aligned}
 F[X; Y, \lambda]_c &=& \{\sum_{\nu} f_\mu X^\mu : \left| f_\mu \right|_\lambda p^{c\cdot \mu} \stackrel{|\mu| \rightarrow \infty}{\rightarrow} 0\}\end{aligned}$$ where $c \cdot \mu = c_1\mu_1 + \cdots + c_n\mu_n$.

Define $\| \; \|_{\lambda, c}$ on $F[X; 