The following lemma is from Qing Liu's "Algebraic Geometry and Arithmetic Curves" p. 61. I don't get line 2-3 of the proof. Why does it suffice to show that $x$ is closed in every affine open subset $V$ that contains it?
$X^0$ is the topological subspace made up of the closed points of $X$, and same for $U^0$.
Let $\{W_i\}_{i\in I}$ be an open cover of a topological space $Y$. Then a subset $Z\subset Y$ is closed (resp. open) iff $W_i\cap Z$ is closed (resp. open) in each $W_i$ for all $i$. Since schemes/varieties are covered by open affine schemes/varieties, the result follows.