Let $A$ be a noetherian ring complete with respect to a principal ideal $(\pi)$: $A\simeq\lim_\leftarrow A/(\pi^n)$. Denote by $X$ the formal scheme $Spf(A)$ and by $X_{n-1}$ the scheme $Spec(A/(\pi^n)$. Let $\cal F$ be a quasi coherent sheaf on $Spf(A)$. Denote by $\cal F_n$ the restriction of $\cal F$ to $X_n$.
Question:
Is it true that $\cal F$ is coherent if and only if $\cal F_0$ is coherent on $X_0$?
Edit: Of course one direction is evident. I think that the other is also true because of the exact sequence, $$0\longrightarrow \cal F_0\otimes (\pi^n)/(\pi^{n-1})\longrightarrow\cal F_n\longrightarrow\cal F_{n-1}\longrightarrow 0$$ and induction on $n$.
But I didn't find the statement in the standard literature, thus I wonder if this argument is wrong for some reasons I do not see (and my misunderstanding of formal geometry).