Hartshorne exercise II.4.5(c) Let $X$ be an integral scheme of finite type over a field $k$, having function field $K$. We say that a valuation of $K/k$ has center $x$ on $X$ if its valuation ring $R$ dominates the local ring $O_{x,X}$.
In part (c) of the question, we are asked to show if every valuation of $K/k$ has a unique center on $X$, then $X$ is proper over $k$.
I am awared that this question was asked in Hartshorne's Exercise II.4.5(c) and I also found a solution in https://en.wikibooks.org/wiki/Solutions_to_Hartshorne%27s_Algebraic_Geometry/Separated_and_Proper_Morphisms#Exercise_II.4.5.
It seems to me that, by using induction, this question come down to showing that for any irreducible reduced closed subscheme, $Z$, of $X$, $Z$ satisfies the conditions on $X$.
That is, I want to show, if every valuation of $K/k$ has a unique center on $X$ and $L$ is the function field of $Z$, then every valuation of $L/k$ has a unique center on $Z$.
How do I show this?
 A: Edit. This argument is currently incomplete: it does not prove that $x$ a center for $T$ implies that $x\in Z$ and $x$ is a center for $R$, just that if a valuation on $Z$ has a center, it is unique. But it seems like this technique should work! If anyone can help fill the missing gaps (or provide another solution), I would welcome hearing from you. I'm leaving this up instead of deleting this in the hopes it will be helpful to someone at some point.

Let $z$ be the generic point of $Z$, $L$ the function field of $Z$, and let $R$ be a valuation ring for $L/k$. Since $L$ is a quotient of $\mathcal{O}_{X,z}$, we can find a ring $S\subset \mathcal{O}_{X,z}$ so that $S/\mathfrak{m}_z=R$. By a standard Zorn's lemma argument, we can find a local ring $T$ maximal with respect to inclusion among local rings containing $S$ so that $T$ is a valuation ring for $K/k$. Then if $z_0\in Z$ is a center for $R$, we have that $z_0\in X$ is a center for $T$, so any valuation ring on $Z$ has a unique center.
This is almost the same as a non-inductive proof, though: given a valuation ring $Q$ with fraction field $M$ and maps $\operatorname{Spec} M\to X$, $\operatorname{Spec} Q\to \operatorname{k}$ with the image of $\operatorname{Spec} M$ being the point $z$, then we can get a valuation ring $R$ for $L$ via taking the intersection of $Q$ with $L\subset M$. From there, unicity/unicity+existence of centers implies separated/proper via the valuative criteria by the argument in the first paragraph.
