For every $n\in\mathbb N$, let $f_n(Z)=\sum_{k\ge0}f^{(n)}_kZ^k\in\mathbb R[[Z]]$ be a formal power series. One assumes that there exists a positive real number $C$ such that for all $(n,k)\in\mathbb N^2$, one has $|f^{(n)}_k|\le C^k$. So, all the power series $f_n(z)$ converges in $]-1/C,C[$. Let $a\in]-1/C,1/C[$. Put $\Gamma_C=\{g(Z)=\sum_{k\ge0}a_kZ^k\in\mathbb R[[Z]]\mid\forall k\in\mathbb N, |a_k|\le C^k\}$. Consider the evaluation map
$\begin{array}{cccc}\sigma_a:&\Gamma_C&\to&\mathbb R\\&f(Z)=\sum_{k\ge0}a_kZ^k&\to& f(a)=\sum_{k=0}^{+\infty}a_ka^k.\end{array}$
The map $\sigma_a$ is a morphism of $\mathbb R$-algebra. And obviously, the $f_n$'s belong to $\Gamma_C$.
I want to prove that if $(f_n)_n$ converges towards $0$ in $\mathbb R[[Z]]$ for the $(Z)$-topology then, $(f_n(a))_n$ converge towards $0$ in $\mathbb R$. Since $(g_n)$ converges towards $0$ in $\mathbb R[[Z]]$, then $\lim_{n\to+\infty}\mathrm{ord}_0(g_n)=+\infty$. But I am stuck here. Any help to conclude the proof is welcome.