= Solution
A <proper morphism> of <schemes> is a <separated morphism> of <finite type> which is a <universally closed morphism>: after every <base change>, the resulting map of underlying <topological spaces> takes closed subsets to closed subsets.
The <valuative criterion for properness>, in the Noetherian setting used here, says that a <finite type morphism> $f:X\to Y$ is a <proper morphism> if and only if the following existence and uniqueness condition holds. For every <valuation ring> $V$, its <fraction field> $K$, and compatible <morphisms of schemes> $a:\operatorname{Spec}K\to X$ and $b:\operatorname{Spec}V\to Y$ with $f\circ a=b|_{\operatorname{Spec}K}$, there is exactly one <morphism of schemes> $h:\operatorname{Spec}V\to X$ such that $h|_{\operatorname{Spec}K}=a$ and $f\circ h=b$. All <valuation rings> are allowed; no discreteness assumption is needed in this formulation. The following arguments also use that each of the three defining conditions for a <proper morphism> is preserved by <base change>.
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