I am trying to understand a theorem on Huber rings and adic spectra. The specific questions is related to a set of lecture notes by Brian Conrad. The overall question, however, is a fact from one of Huber's paper (linked below) that analytic points of continuous valuation spectra are those containined in a neighborhood defined by a Tate ring.
Let $A$ be a Huber ring and $\mathrm{Cont}(A)$ be its continuous valuation spectrum. Suppose that $(A_0,I)$ is a ring/ideal of definition for $A$. Write $T = \{f_1,\dotsc,f_n\}$ for a set of $A_0$-generators of the ideal $I$. Then we can form rational localizations $A(T/f_i)$ for each $i=1,\dotsc,n$. These are $A$-algebras that are initial among Huber $A$-algebras $\varphi: A \rightarrow B$ where $\varphi(f_i) \in B^\times$ and $\varphi(f_j)/\varphi(f_i)$ is power-bounded in $B$ for all $j=1,\dotsc,n$. The underlying $A$-algebra is $A[1/f_i]$ and the topology is the one defined by a neighborhood basis of $0$ being $f_i^dA_0[f_j/f_i \; j=1,2,\dotsc,n]$ for $d=0,1,2,\dotsc$. The Huber ring $A(T/f_i)$ is an example of a Tate ring.
The statement I want to understand is:
The image of $\mathrm{Cont}(A(T/f_i)) \rightarrow \mathrm{Cont}(A)$ is exactly $U_{\mathrm{cont}}(T/f_i) = \{v \in \mathrm{Cont}(A) \mid v(f_j)\leq v(f_i) \neq 0 \;\; j=1,\dotsc,n\}$
This is the statement of Proposition 8.3.6 in Conrad's notes.
The challenge I am having is in the argument for the proposition. There, it is claimed that if $v$ is a valuation on $A(T/f_i)$ then the continuity of $v$ is equivalent to $v(f_j/f_i) \leq 1$ (where the quotient $f_j/f_i$ exists in $A(T/f_i)$). Why? (Is it right?)
The universal property of $A(T/f_i)$ says $f_j/f_i$ is power-bounded, but one of the great counter-intuitive parts of Huber's valuation theory is that power-bounded elements need not satisfy $v(-)\leq 1$ when $v$ has rank $>1$. In other words, if $v$ were a rank one valuation then I understand the deduction, but I don't see the implication if $v$ were to have higher rank. (And yet I haven't constructed a counter-example.)
Many people have studied Conrad's notes, so I'm hoping someone here can help.
Huber, Roland, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217, No. 4, 513-551 (1994). ZBL0814.14024.