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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 $$M_F(f,r)=\lim_{n\to+\infty}M_{\mathbb C_p}(f,r_n)?$$

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  • $\begingroup$ For $f$ a $p$-adic Laurent series and $r$ in the open disk of convergence, reducing $p^m f(z)$ modulo $(p^l,z^N)$ don't you get that $\inf_{v(z) \ge r} v(f(z)) = \inf_n v(a_n) + rn$ $\endgroup$ – reuns Oct 25 '19 at 20:31
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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.

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