= Solution
Let $R$ be a <discrete valuation ring> with <uniformizer> $\pi$, <fraction field> $K$, and normalized discrete valuation $\nu:K^\times\to\mathbb Z$. A map $\operatorname{Spec}K\to\mathbb P^n_{\mathbb Z}$ is a <projective point>
$$
[a_0:\cdots:a_n],\qquad a_i\in K,
$$
with at least one nonzero coordinate. Put $m=\min_{a_i\ne0}\nu(a_i)$ and set $b_i=\pi^{-m}a_i$. Then every $b_i$ lies in $R$, and at least one $b_j$ is a unit.
On the standard affine chart $U_j$ of <projective space>, the ratios $b_i/b_j$ all lie in $R$. They therefore define a map $\operatorname{Spec}R\to U_j\subseteq\mathbb P^n_{\mathbb Z}$ whose generic restriction is the original point. This proves existence in the DVR case. The assumed separatedness of $\mathbb P^n_{\mathbb Z}\to\operatorname{Spec}\mathbb Z$, through the <valuative criterion for separatedness>, gives uniqueness. Thus the morphism satisfies the requested DVR form of the <valuative criterion for properness>.
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