Under AC, effective divisors on normal proper curves give finite subschemes of the same degree

For every $x\in S$ there exist an affine open subset $U_x=\mathrm{Spec}{A}_x$ containing $x$ and an element $f_x\in A_x$ such that $U_x\cap S=\{x\}$ and the only zero of $f_x$ in $U_x$ is $x$, with ${\mathrm{ord}}_x(f_x)=1$. Indeed, fix $x\in S$ and choose an affine open $U=\mathrm{Spec}A\ni x$ (of $C$, by [F6]). By [F1] and [F2] the local ring ${\mathcal{O}}_{C,x}=A_{{\mathfrak{m}}_x}$ is a discrete valuation ring with fraction field $K$; choose $\pi \in {\mathcal{O}}_{C,x}$ with $v_x(\pi )=1$ and write $\pi =b\mathrm{/}s$ with $b\in A$ and $s\mathrm{\notin }{\mathfrak{m}}_x$. Then $v_x(b)=v_x(\pi )+v_x(s)=1$. By [F3] the principal divisor ${\mathrm{div}}_W(b)$ has finite support, so $T=(\mathrm{supp}{\mathrm{div}}_W(b)\cup S)\setminus \{x\}$ is a finite set of closed points not containing $x$; being a finite union of singleton closed sets, $T$ is closed, so $W=U\setminus T$ is an open neighbourhood of $x$. Choose an affine open $U_x=\mathrm{Spec}{A}_x$ with $x\in U_x\subseteq W$ ([F6]) and put $f_x=b{\mathrm{\mid }}_{U_x}$. Then $U_x\cap S=\{x\}$, and for every closed point $y\in U_x$ with $y\mathrm{\ne }x$ we have $y\mathrm{\notin }\mathrm{supp}{\mathrm{div}}_W(b)$, so ${\mathrm{ord}}_y(f_x)=0$ and $f_x$ does not vanish at $y$. At $x$ we have ${\mathrm{ord}}_x(f_x)=1$ by construction, so the only zero of $f_x$ in $U_x$ is $x$.

For every $x\in S$ the principal ideal $(f_x)\subseteq A_x$ is the maximal ideal ${\mathfrak{m}}_x$ of $x$, so $A_x\mathrm{/}(f_x)\cong \kappa (x)$. First note that $A_x$ is a domain with fraction field $K$ by [F2], so the quotient field of fractions used below is legitimate. Let $g\in {\mathfrak{m}}_x$; we show $g\in (f_x)$. The element $h:=g\mathrm{/}{f}_x\in K$ satisfies $h\in (A_x{)}_{\mathfrak{m}}$ for every maximal ideal $\mathfrak{m}\subseteq A_x$: if $\mathfrak{m}={\mathfrak{m}}_x$ then $v_x(g)\ge 1=v_x(f_x)$, while if $\mathfrak{m}\mathrm{\ne }{\mathfrak{m}}_x$ then $g\in A_x$ gives $v_{\mathfrak{m}}(g)\ge 0$ and $f_x\mathrm{\notin }\mathfrak{m}$ gives $v_{\mathfrak{m}}(f_x)=0$, the latter because $\mathfrak{m}$ corresponds to a closed point $y\in U_x$ with $y\mathrm{\ne }x$ and $f_x$ does not vanish at $y$ (step 1.1). We now use the standard fact that a domain equals the intersection of its localisations at maximal ideals: if $h\in (A_x{)}_{\mathfrak{m}}$ for every maximal ideal $\mathfrak{m}$, then $h\in A_x$. To prove it, write $h=a\mathrm{/}b$ with $a,b\in A_x$, $b\mathrm{\ne }0$, and put $I=\{c\in A_x:ch\in A_x\}$, an ideal containing $b$; if $I\mathrm{\ne }{A}_x$, then by [F5] there is a maximal ideal $\mathfrak{m}\supseteq I$, but $h\in (A_x{)}_{\mathfrak{m}}$ means $h=a^{\mathrm{\prime }}\mathrm{/}s$ with $s\mathrm{\notin }\mathfrak{m}$, whence $sh=a^{\mathrm{\prime }}\in A_x$ and $s\in I\subseteq \mathfrak{m}$, a contradiction. Hence $I=A_x$ and $h\in A_x$. Therefore $g=f_{x}h\in (f_x)$, so ${\mathfrak{m}}_x\subseteq (f_x)$; the reverse inclusion holds because $v_x(f_x)=1>0$ gives $f_x\in {\mathfrak{m}}_x$. Thus $(f_x)={\mathfrak{m}}_x$ and $A_x\mathrm{/}(f_x)=A_x\mathrm{/}{\mathfrak{m}}_x=\kappa (x)$.

The equations $f_x^{n_x}$ on $U_x$ for $x\in S$, together with the equation $1$ on the open complement $U_0=C\setminus S$, form an effective Cartier divisor on $C$; its associated closed subscheme $Z_D$ satisfies $Z_D\cap U_x=\mathrm{Spec}(A_x\mathrm{/}(f_x^{n_x}))$ for $x\in S$ and $Z_D\cap U_0=\mathrm{\varnothing }$, so its support is $S$. Moreover the local equation on $U_x$ has order $n_x$ at $x$ and order $0$ at every other point of $U_x$. The sets $U_x$ ($x\in S$) together with $U_0$ cover $C$: a point of $S$ lies in its own $U_x$, and a point outside $S$ lies in $U_0$. Each equation is a regular section: $f_x^{n_x}\mathrm{\ne }0$ in the domain $A_x$ when $n_x>0$, and $1$ is a unit. On an overlap $U_x\cap U_y$ with $x\mathrm{\ne }y$ the quotient $f_x^{n_x}\mathrm{/}{f}_y^{n_y}$ is a unit, because $U_x$ contains no point of $S$ other than $x$ and $f_x$ vanishes only at $x$ in $U_x$ (step 1.1), so $f_x$ is a unit on $U_x\cap U_y$, and likewise for $f_y$; on $U_x\cap U_0$ the same argument shows that $f_x^{n_x}$ is a unit. Hence the data glue to a Cartier divisor $D^{\mathrm{\prime }}$ by [F4], and $D^{\mathrm{\prime }}$ is effective because all equations are regular. By [F4] and [F6] its associated closed subscheme has $I_{D^{\mathrm{\prime }}}{\mathrm{\mid }}_{U_x}=f_x^{n_x}{\mathcal{O}}_{U_x}$ and $I_{D^{\mathrm{\prime }}}{\mathrm{\mid }}_{U_0}={\mathcal{O}}_{U_0}$, so $Z_D\cap U_x=\mathrm{Spec}(A_x\mathrm{/}(f_x^{n_x}))$ and $Z_D\cap U_0=\mathrm{\varnothing }$. The order of the local equation $f_x^{n_x}$ at $x$ is $n_x{\mathrm{ord}}_x(f_x)=n_x$, and at every other point of $U_x$ it is $0$; on $U_0$ the equation $1$ has order $0$ everywhere.

For every $x\in S$ and every integer $n\ge 0$ one has ${\dim }_k{A}_x\mathrm{/}(f_x^n)=n[\kappa (x):k]$; in particular $A_x\mathrm{/}(f_x^n)$ is a finite-dimensional $k$-vector space. Since $A_x$ is a domain and $f_x\mathrm{\ne }0$, multiplication by $f_x^j$ induces, for each $j\ge 0$, an isomorphism of $A_x$-modules $A_x\mathrm{/}(f_x)\to (f_x^j)\mathrm{/}(f_x^{j+1})$, $a\mapsto a{f}_x^j$: it is surjective, and $a{f}_x^j\in (f_x^{j+1})$ implies $a\in (f_x)$ because $A_x$ is a domain. The chain $A_x\mathrm{/}(f_x^n)\supseteq (f_x)\mathrm{/}(f_x^n)\supseteq (f_x^2)\mathrm{/}(f_x^n)\supseteq \cdots \supseteq (f_x^n)\mathrm{/}(f_x^n)=0$ therefore has $n$ successive quotients isomorphic to $A_x\mathrm{/}(f_x)$, each of $k$-dimension $[\kappa (x):k]$ by step 2.1 and [F1]. Since $k$-dimension is additive in such finite filtrations, ${\dim }_k{A}_x\mathrm{/}(f_x^n)=n[\kappa (x):k]$.

The scheme $Z_D$ is finite over $k$ and ${\dim }_k\mathrm{\Gamma }(Z_D,{\mathcal{O}}_{Z_D})={\sum }_{x\in S}{n}_x[\kappa (x):k]={\mathrm{deg}}_{k}D$. By step 2.2 the subschemes $Z_D\cap U_x$ for $x\in S$ form an open cover of $Z_D$ with pairwise empty intersections, so the sheaf axioms identify $\mathrm{\Gamma }(Z_D,{\mathcal{O}}_{Z_D})={\prod }_{x\in S}{A}_x\mathrm{/}(f_x^{n_x})$ as $k$-algebras and as $k$-vector spaces; when $S$ is empty this is the zero ring and $Z_D=\mathrm{\varnothing }$. Each factor is finite-dimensional over $k$ by step 3.1, so the product is a finite-dimensional $k$-algebra, of dimension ${\sum }_{x\in S}{n}_x[\kappa (x):k]$ by [F6]; this equals ${\mathrm{deg}}_{k}D$ by [F1], because the terms with $n_i=0$ contribute nothing. Being the spectrum of a finite-dimensional $k$-algebra, $Z_D$ is finite over $k$; more precisely the product decomposition of [F6] exhibits $Z_D$ as the disjoint union of the affine schemes $\mathrm{Spec}(A_x\mathrm{/}(f_x^{n_x}))$.

Conclusion. Every effective divisor $D={\sum }_i{n}_i[x_i]$ with $n_i\ge 0$ on a normal proper integral curve $C$ over $k$ determines an effective Cartier divisor whose vanishing subscheme $Z_D$ is finite over $k$, supported on the $x_i$ with $n_i>0$, of $k$-length ${\dim }_k\mathrm{\Gamma }(Z_D,{\mathcal{O}}_{Z_D})={\sum }_i{n}_i[\kappa (x_i):k]={\mathrm{deg}}_{k}D$. The Axiom of Choice is used exactly as declared, through [F5] in the intersection step 2.1, and it also supplies the Dependent Choice used for the finiteness of ${\mathrm{div}}_W(b)$ in [F3]; the remaining steps are choice-free.