[proofplan] The proof uses a finite Stein cover of the compact manifold and a functional-analytic compactness argument. Coherence gives finite free resolutions on coordinate polydiscs, so the Čech cochain spaces are Fréchet spaces of holomorphic sections. Restriction maps between slightly smaller polydiscs are compact operators by Montel's theorem. A finite complex with compact perturbations then has finite-dimensional cohomology. [/proofplan] [step:Choose a finite adapted cover] By compactness, choose finitely many coordinate polydiscs $U_i$ covering $X$, together with slightly larger polydiscs $U_i'$ whose closures contain $U_i$. Refining if necessary, every finite intersection that occurs is Stein. Since $\mathcal{F}$ is coherent, on each such intersection it admits a finite presentation by free analytic sheaves. [/step] [step:Represent cochains by Fréchet spaces] For the finite cover, the Čech cochain group \begin{align*} C^q(\mathcal{U},\mathcal{F})=\prod_{i_0<\cdots<i_q}\mathcal{F}(U_{i_0\cdots i_q}) \end{align*} is a finite product of spaces of holomorphic sections. With the topology of uniform convergence on compact subsets, these are Fréchet spaces. The Čech differential is a continuous linear map between Fréchet spaces. [/step] [step:Use compactness of restriction maps] Restriction from a larger polydisc to a relatively compact smaller polydisc is compact on spaces of holomorphic functions by Montel's theorem: bounded families are normal, so restrictions have convergent subsequences. The same holds for coherent sheaves after choosing finite local presentations. Thus the Čech complex can be compared with a complex built on the larger cover by compact restriction operators. [/step] [step:Apply the Cartan-Serre compact operator lemma] The Cartan-Serre lemma says that in this situation, the cohomology of the finite Fréchet complex is finite-dimensional: compactness of the comparison maps prevents infinite-dimensional closed obstruction spaces from persisting. Applying this lemma to the Čech complex gives finite-dimensional Čech cohomology. Since the cover is acyclic for coherent sheaves on the relevant intersections, this Čech cohomology equals sheaf cohomology. Therefore $H^q(X,\mathcal{F})$ is finite-dimensional for every $q\ge0$. [/step]