I'm studying algebraic geometry by following Mumford's Red Book, but recently I found Qing Liu's book "Algebraic Geometry and Arithmetic Curves", which seems to be more careful and detailed in its proofs $-$ what I prefer, because I'm novice in this subject.
However, in Theorem 3.25 of Liu's book (pg 107) $-$ whose immediate corollary corresponds to Proposition 3 of Section 7 in Chapter 2 of Mumford's book $-$ I am just as stuck as when I was following Mumford's book.
The theorem is the following: (Theorem 3.25) Let $X$ be a proper scheme over a valuation ring $\mathcal{O}_K$, and let $K=\text{Frac}(\mathcal{O}_K)$. Then the canonical map $X(\mathcal{O}_K)\rightarrow X_K(K)$ is bijective.
This is the proof:
Essentially I have "two" questions.
The first question: How this proof works? Although I feel bad for asking for a walkthrough to this proof, I can't "connect" its many statements in an one whole argument, at least no with the amount of knowledge I've got so far. In some sense, this question is the same as Proof of $X(\mathcal{O}_K)\simeq X_K(K)$ , but without any "ok" on the part of surjectivity.
The second question: if one supposes $X$ to be affine, then this theorem can be proved by using some analogous result in commutative algebra in the context of tensor products?
Thank you!