Let $F$ be the spherically complete extension of $\mathbb C\_p$ and $(a\_n)\_{n\in\mathbb N}$ be a sequence of $\mathbb C\_p$ such that for all $r\in\mathbb R$, one has $$\lim\_{n\to+\infty}|a\_n|\_pr^n=0.$$ Put 
$f(z)=\sum\_{n\ge0}a\_nz^n$. For a non-negative real number $r$ define the *modulus of growth in $F$* by $$M\_F(f,r)=\inf\{v\_p(f(z))\mid z\in F\text{ with }v\_p(z)\ge r\}.$$ For a non-negative rational number $r$ define the *modulus of growth in $\mathbb C\_p$* by $$M\_{\mathbb C\_p}(f,r)=\inf\{v\_p(f(z))\mid z\in\mathbb C\_p\text{ with }v\_p(z)\ge r\}.$$ My question:
for every positive real number $r$, can one find a sequence $(r\_n)\_{n\in\mathbb N}$ of rational numbers such that $$M\_F(f,r)=\lim\_{n\to+\infty}M\_{\mathbb C\_p}(f,r\_n)?$$ 



Modulus of growth in $p$-adic spherically complete field of $\mathbb C_p$
It is known that for an entire function $f(x)=\sum\_{n\geq 0} a\_nx^n\in{\mathbb C}\_p[[x]]$, the function
\begin{equation\*}
r\longrightarrow M(f,r)=\inf\_{n\geq 0} v(a\_n)+n r=\inf\_{v(x)x\geq r} v(f(x))
\end{equation\*}
is continuous and piecewise affine, cf. A. Robert,\emph{A course in $p$-adic analysis}, Springer Verlag GTM 198. 


