Let $X$ be a scheme and $\mathcal{L}$ be an ample invertible sheaf on $X$, then lemma 28.26.7 in stacks project says that $X$ is separated.
The proof uses the valuative criterion and goes as follows. Concretely, let $A$ be a valuation ring with fraction field $K$ and consider two morphisms $f, g: \mathrm{Spec}A\to X$ such that the two compositions $\mathrm{Spec}K\to\mathrm{Spec}A\to X$ agree. As $A$ is local, there exists $p, q\geq1$, $s\in\Gamma(X, \mathcal{L}^{\otimes p})$ and $t\in\Gamma(X, \mathcal{L}^{\otimes q})$ such that $X_s$ and $X_t$ are affine, $f(\mathrm{Spec}A)\subseteq X_s$ and $g(\mathrm{Spec}A)\subseteq X_t$.
Question: The existence of $s$ and $t$ such that $X_s$ and $X_t$ are affine follows from the definition of ample sheaf given by definition 28.26.1, but I don't know why $f(\mathrm{Spec}A)\subseteq X_s$ and $g(\mathrm{Spec}A)\subseteq X_t$.
Thanks in advance!
That is because $A$ is local. Let $x\in X$ be the image of the closed point of $\operatorname{Spec} A$, then you have a morphism on stalks $\mathcal O_{X,x}\to A$. Say $B$ is the coordinate ring of $X_s$(which is a neighborhood of $x$), then $\mathcal O_{X,x}$ is a localisation of $B$, and the morphism $\operatorname{Spec} A \to X_s$ is given by $B\to \mathcal O_{X,x} \to A$.