Every smooth proper curve admits a projective embedding

The curve is integral, proper and of chain dimension one. It is of finite type over $k$ by the curve definition, and $\mathrm{Spec}k$ is Noetherian. Thus $C\to \mathrm{Spec}k$ is a proper finite-type morphism, and $C$ is a proper curve in the sense required by [F2].

The function field has transcendence degree one. Chain dimension one gives a strict chain $Z_0⊊Z_1$ of nonempty irreducible closed subsets of $C$. Since $C$ is itself irreducible and closed, $Z_1=C$; otherwise $Z_0⊊Z_1⊊C$ would be a chain of length two. Choose $p\in Z_0$. As $Z_0⊊C$, the point $p$ is not the generic point, so it is closed by [F2]. Choose an affine open $U=\mathrm{Spec}A$ containing $p$, and let $\mathfrak{m}$ be its maximal ideal. By [F5], $\dim A_{\mathfrak{m}}=\dim {\mathcal{O}}_{C,p}=1$, so $\dim A\ge 1$. Any strict chain of irreducible closed subsets of $U$ remains strict after taking closures in $C$: each closed subset is recovered by intersecting its closure with $U$. Hence $\dim A=\dim U\le \dim C=1$, and $\dim A=1$. Applying [F4] and [F3] gives ${\mathrm{trdeg}}_{k}k(C)=1$.

The curve is normal in the sense used in [F7]. Every point is either the generic point or closed [F2]. The generic local ring is $k(C)$, a field [F3]. At each closed point the local ring is a DVR [F5], hence an integrally closed domain [F6].

Choose a transcendental element $f\in k(C)$, which exists because ${\mathrm{trdeg}}_{k}k(C)=1$ [F4, step 1.2]. It is nonzero, and step 1.3 verifies normality, so [F8] provides a finite locally free morphism ${\phi }_f\text{\hspace{0.17em}⁣}:C\to {\mathbf{P}}_k^1$.

The sheaf ${\mathcal{O}}_{{\mathbf{P}}_k^1}(1)$ is ample by [F9]. Since ${\phi }_f$ is finite, [F10] makes its pullback $L:={\phi }_f^{\ast }{\mathcal{O}}_{{\mathbf{P}}_k^1}(1)$ an ample invertible sheaf on $C$.

The base $\mathrm{Spec}k$ is Noetherian; the structure morphism of $C$ is proper of finite type [F2, step 1.1]; and $L$ is ample [F10, step 3.1]. The ample-powers theorem [F11] therefore gives a positive integer d such that $L^{\otimes d}$ is closed H-very ample relative to $\mathrm{Spec}k$.

By [F12], this closed H-very ample sheaf yields an integer $N\ge 0$ and a closed immersion $i\text{\hspace{0.17em}⁣}:C\to {\mathbf{P}}_k^N$. Its composite with ${\mathbf{P}}_k^N\to \mathrm{Spec}k$ is the structure morphism, proving projectivity in the stated H-projective convention. This argument uses no Serre duality, Riemann-Roch or residue theory; the Axiom of Choice is inherited through the declared curve, rational-map, projective-space and ample-power suppliers.