Let $(R, M)$ be a valuation ring with algebraically closed fraction field $k$. Let $L = R/M$ be the residue field of $R$; it follows that $L$ is algebraically closed. I would like to understand the following: Suppose $Z$ is an irreducible closed subscheme of $\mathbb{P}^n_k$ defined by the homogeneous ideal $A \subset k[X_0, ..., X_n]$. Let $B = \pi\{ A \cap R[X_0, ..., X_n] \}$ where $\pi$ is the quotient modulo $M$. Let $W \subset \mathbb{P}^n_L$ be defined by $B$. Suppose $[\bar{a_0}:...: \bar{a_n}]$ is a closed point of $W$. Then I would like to know how one can prove that there exit $b_0, ..., b_n \in R$ (not all in $M$) such that $b_j + M = \bar{a_j}$ for each $0 \leq j \leq n$ and $[b_0:...: b_n]$ is a closed point of $Z$.

I am asking this question because I would like to understand the proof of the following result.
In Mumford's *'Red book of schemes'*, Theorem 1 Chapter II Section 8, he proves:
For all closed subsets $Z \subset \mathbb{P}^n_k$, there is a unique closed subset
$W \subset \mathbb{P}^n_L$ such that
$$
\rho(Z(k)) = W(L),
$$
where $\rho: \mathbb{P}^n(k) \to \mathbb{P}^n(L)$.

I am having similar issues with the proof as in this MathSE question. If someone could provide an alternative reference for this result or an explanation for the statement in the first paragraph, it would be appreciated. Thank you