$\mathcal{O}_X(U)=\cap_{x\in U}\mathcal{O}_{X,x}$ in integral schemes

Question

Let $X$ be an integral scheme with a generic point $\xi$.

I understand:

1. For every affine open $V\subset X$, $\text{Frac}\mathcal{O}_X(V)\cong \mathcal{O}_{X,\xi}$
2. For every open $U\subset X$ and $x\in X$, $\mathcal{O}_X(U)\rightarrow\mathcal{O}_{X,x}$ and $\mathcal{O}_{X,x}\rightarrow\mathcal{O}_{X,\xi}$ are injective.
3. For every affine open $U\subset X$, $\mathcal{O}_X(U)=\cap_{x\in U}\mathcal{O}_{X,x}$ (regarding both sides as subrings of $\mathcal{O}_{X,\xi}$)

But in Qing Liu's "Algebraic Geometry and Arithmetic Curves" Prop. 4.18 on page 65, it is claimed that (3) is true for every open $U$ (not necessarily affine). The given reasoning is "by covering $U$ with open affine subsets".

Up until now I managed to fill in the details when such reasoning was given, but not here. What is the accurate explanation of the reduction to the affine case here?

Answer 1

$\mathcal O_X(U) \subset \cap_{x \in U} \mathcal O_{X,x}$ is obvious from 2.

For the other direction let $U=\bigcup_i U_i$ be an affine open cover and let $s \in \cap_{x \in U} \mathcal O_{X,x} \subset \cap_{x \in U_i} \mathcal O_{X,x}$.

By the result for affine opens, you get $s \in \mathcal O_X(U_i)$ for all $i$. Clearly $s$ agrees with itself on the intersections $U_i \cap U_j$, i.e. by the sheaf axiom it extends to a section on $\mathcal O_X(U)$.