Let $R$ be a discrete valuation ring, $\{A_i\}_{i \in I}$ be a direct system of $R$-algebras and $A$ the limit of the system. Let $X$ be a noetherian projective scheme over $\mathrm{Spec}(R)$. Suppose that there exists an open subset $U \subset X$ and a morphism $f:\mathrm{Spec}(A) \to U$. Then, is it true that for all but finitely many $i \in I$, there exists a morphism $f_i:\mathrm{Spec}(A_i) \to U$ such that its composition with the natural morphism $\mathrm{Spec}(A) \to \mathrm{Spec}(A_i)$ is simply $f$?
The motivation of the question comes from a similar result which holds true for sequences of closed points on topological spaces. I do not know if this question is obvious but was unable to find a reference or counterexample.
