[proofplan] Serre's proof reduces coherent sheaves on a projective variety to twists of the structure sheaf. For large twists, Serre vanishing and global generation convert sheaves into finite graded modules. Analytic and algebraic cohomology agree for the twisting sheaves on projective space, and exact sequences propagate the comparison to all coherent sheaves. Essential surjectivity follows by reconstructing an analytic coherent sheaf from the finite graded module of its twisted global sections. [/proofplan] [step:Prove the comparison for projective space twists] On $\mathbb{P}^N_\mathbb{C}$, the algebraic global sections of $\mathcal{O}(m)$ are homogeneous polynomials of degree $m$. The analytic global sections are the same by compactness and the classical fact that a holomorphic section of $\mathcal{O}(m)$ is represented by homogeneous holomorphic functions, hence homogeneous polynomials. The higher cohomology of $\mathcal{O}(m)$ is computed identically on both sides by the standard Čech cover by affine charts. Thus the comparison is true for finite direct sums of twisting sheaves. [/step] [step:Resolve coherent algebraic sheaves by twists] For a coherent algebraic sheaf $\mathcal{F}$ on projective $X$, Serre's theorem gives $m\gg0$ such that $\mathcal{F}(m)$ is globally generated and has vanishing higher algebraic cohomology. Therefore there is a surjection from a finite direct sum of $\mathcal{O}_X(-m)$ onto $\mathcal{F}$. Iterating on the kernel gives a finite resolution by sums of twists in the range needed to compute cohomology. [/step] [step:Propagate cohomology comparison] Analytification is exact on coherent algebraic sheaves and sends $\mathcal{O}_X(m)$ to $\mathcal{O}_{X^{\mathrm{an}}}(m)$. Applying algebraic and analytic cohomology to the same finite resolutions gives parallel long exact sequences. Since the comparison maps are isomorphisms for the twists, the five lemma gives \begin{align*} H^q(X,\mathcal{F})\cong H^q(X^{\mathrm{an}},\mathcal{F}^{\mathrm{an}}) \end{align*} for all $q\ge0$. [/step] [step:Recover analytic sheaves algebraically] Let $\mathcal{G}$ be a coherent analytic sheaf on $X^{\mathrm{an}}$. For large $m$, Cartan-Serre finiteness and analytic global generation give a finite-dimensional space of sections of $\mathcal{G}(m)$ and a surjection from a finite sum of $\mathcal{O}_{X^{\mathrm{an}}}(-m)$ onto $\mathcal{G}$. The kernels are coherent analytic sheaves, so repeating gives a finite analytic presentation by twists. The comparison for Hom groups between twists identifies the matrices in this presentation with algebraic matrices. Thus the presentation descends to a coherent algebraic sheaf whose analytification is $\mathcal{G}$. The same Hom comparison gives uniqueness and full faithfulness. [/step]