A meromorphic unit has locally finite nonzero order support

Local fraction representation. For every point $x\in X$ there are an affine open $U=\mathrm{Spec}A$ containing $x$ and elements $a,s\in A$ such that $f{\mathrm{\mid }}_U$ is the image of $a\mathrm{/}s\in {\mathcal{P}}_X(U)$ under the sheafification map. Since ${\mathcal{K}}_X=a{\mathcal{P}}_X$, [F3] gives, for the section $f$ around $x$, an open neighbourhood $V$ of $x$ and an element $t\in {\mathcal{P}}_X(V)$ whose image in ${\mathcal{K}}_X(V)$ is $f{\mathrm{\mid }}_V$. The affine open subschemes form a basis, so there is an affine open $U=\mathrm{Spec}A\subseteq V$ containing $x$; restricting $t$ to $U$ gives an element $a\mathrm{/}s$ of ${\mathcal{P}}_X(U)=S_X(U{)}^{-1}A$ whose image in ${\mathcal{K}}_X(U)$ is $f{\mathrm{\mid }}_U$. Only the local representation is used, and no choice is made from an infinite family.

Finitely many candidates over the chart. For $U=\mathrm{Spec}A$ and $a\mathrm{/}s$ as in 1.1, only finitely many prime divisors $Z$ with $Z\cap U\mathrm{\ne }\mathrm{\varnothing }$ satisfy ${\mathrm{ord}}_Z(f)\mathrm{\ne }0$: each such $Z$ corresponds to a prime ideal of $A$ that is minimal over $(a)$ or over $(s)$. Let $Z$ be a prime divisor with $Z\cap U\mathrm{\ne }\mathrm{\varnothing }$. The scheme $Z$ is integral, so by [F8] its generic point $\xi$ lies in the nonempty open subset $Z\cap U$ of $Z$; hence $\xi \in U$, and $\xi$ corresponds to a prime $\mathfrak{p}\subseteq A$ with ${\mathcal{O}}_{X,\xi }=A_{\mathfrak{p}}$ and, by [F4], $\dim A_{\mathfrak{p}}=\dim {\mathcal{O}}_{X,\xi }=1$. Write $a_{\xi }$ and $s_{\xi }$ for the images of $a$ and $s$ in $A_{\mathfrak{p}}$. Since $s\in S_X(U)$, its germ $s_{\xi }$ is a nonzerodivisor of the domain $A_{\mathfrak{p}}$, so $s_{\xi }\mathrm{\ne }0$; the germ of the class of $a\mathrm{/}s$ at $\xi$ is the fraction $a_{\xi }\mathrm{/}{s}_{\xi }$. By [F4] and [F2] the stalk ${\mathcal{K}}_{X,\xi }$ is the fraction field of ${\mathcal{O}}_{X,\xi }=A_{\mathfrak{p}}$, the germ of $f$ there is the image of $a\mathrm{/}s$, and $f_{\xi }\in K(X_i{)}^{\times }$ is a unit of that field; under the identification with $a_{\xi }\mathrm{/}{s}_{\xi }$ this gives $a_{\xi }\mathrm{/}{s}_{\xi }\mathrm{\ne }0$, hence $a_{\xi }\mathrm{\ne }0$. With $v_{\xi }$ the normalised valuation we thus have ${\mathrm{ord}}_Z(f)=v_{\xi }\text{\hspace{0.17em}⁣}\left(\frac{a_{\xi }}{s_{\xi }}\right)=v_{\xi }(a_{\xi })-v_{\xi }(s_{\xi }),$ and $v_{\xi }(a_{\xi })\ge 0$ because $a_{\xi }\in A_{\mathfrak{p}}$. Suppose first that $v_{\xi }(a_{\xi })>0$, so that $a\in \mathfrak{p}$ by [F9]. If $\mathfrak{q}$ is a prime with $(a)\subseteq \mathfrak{q}\subseteq \mathfrak{p}$, then $a_{\xi }\mathrm{\ne }0$ lies in $\mathfrak{q}{A}_{\mathfrak{p}}$, so $0⊊\mathfrak{q}{A}_{\mathfrak{p}}\subseteq \mathfrak{p}{A}_{\mathfrak{p}}$; in the one-dimensional local domain $A_{\mathfrak{p}}$ every nonzero prime is the maximal ideal, so $\mathfrak{q}{A}_{\mathfrak{p}}=\mathfrak{p}{A}_{\mathfrak{p}}$, and contracting gives $\mathfrak{q}=\mathfrak{p}$. Hence $\mathfrak{p}$ is minimal over $(a)$. Otherwise $v_{\xi }(a_{\xi })=0$, and ${\mathrm{ord}}_Z(f)\mathrm{\ne }0$ forces $v_{\xi }(s_{\xi })\mathrm{\ne }0$, so $s\in \mathfrak{p}$ by [F9]; the same argument, now with $s_{\xi }\mathrm{\ne }0$, shows that $\mathfrak{p}$ is minimal over $(s)$. Thus every such $\mathfrak{p}$ is a minimal prime of one of the Noetherian quotient rings $A\mathrm{/}(a)$ or $A\mathrm{/}(s)$, of which there are finitely many by [F6] and [F7]. Finally the assignment $Z\mapsto \mathfrak{p}$ is injective, because distinct prime divisors have distinct generic points and the prime of $A$ determines the point of $U$. This gives the finiteness asserted.

Local finiteness. For every point $x$ of $X$, the affine open neighbourhood $U$ produced in 1.1 meets only finitely many prime divisors $Z$ with ${\mathrm{ord}}_Z(f)\mathrm{\ne }0$, by 1.2. Hence the family of such $Z$ is locally finite.