$\newcommand{\nrm}[1]{\left\|#1\right\|}
\newcommand{\nr}{\nrm{-}}$
Hello community.
I would like to ask whether some of You knows Literature or proof on the following result:
Let $K$ be a non-Archimedean valued field and let $A,B$ be $K$-algebras of topologically finite type in the sense that it is topologically isomorphic to $\mathbb{T}_n/J$ where $\mathbb{T}_n$ is the Tate Algebra and $J\subseteq \mathbb{T}_n$ is an ideal.
(I think that in Literature this is called an affinoid Algebra, but we used "topologically of finite type" in lecture.)
We defined the spectral norm of $A$ by $\nrm{a}_{\operatorname{spec}}:=\operatorname{lim}_{n\to \infty}\nrm{a^n}_{\operatorname{res}}^{\frac{1}{n}}$ where $\nr_{\operatorname{res}}$ is the residue norm and the $\operatorname{sup}$-norm which is defined by $\nrm{a}_{\operatorname{sup}}:=\sup_{\mathfrak{m}\subseteq A}|f\mod \mathfrak{m}|$.
Then we proved, that these two norms are equal, i.e. that
$\nr_{\operatorname{sup}}=\nr_{\operatorname{spec}} (\star)$.
In the proof we used the following Lemma: Let $A=B/I$. If $(\star)$ holds for all $K$-algebras $C$ of topologically finite type of Krull dimension $\operatorname{dim}(C)<\operatorname{dim}(B)$, then it holds for $A$.
My Question is Where I can find either a comprehensible proof of $(\star)$ or a proof of the Lemma in Literature, because the proof from Lecture is very confusing for me. I couldn't find this result in BGR nor in any other scan/paper/book etc. on this topic.
Thank You for Your help and efforts,
SDIGR