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Fix a coherent sheaf $F$ and an integer $k>M$. Set $K_0=F$. Repeatedly apply a to construct short exact sequences of sheaveswhere each $G_j$ is a coherent sheaf and a locally free sheaf. Each kernel sheaf $K_{j+1}$ is a coherent sheaf because $X$ is a noetherian scheme. Thus the construction can be repeated as many times as needed; it does not require the kernel sheaves themselves to be locally free sheaves.

For every integer $q>M$, the long exact sequence in sheaf cohomology containsBoth outer groups vanish by the hypothesis, since $q$ and $q+1$ exceed $M$. Hence ${\delta }_j$ is an isomorphism. Iterating this dimension shifting in sheaf cohomology gives

Choose $s\ge 0$ with $k+s>N$, where $N$ is the uniform bound from b. The coherent sheaf $K_s$ is a quasi-coherent sheaf, so b makes the last group zero. ThereforeThe argument uses a finite initial segment of a locally free resolution. There is no need to assume that every coherent sheaf has a finite locally free resolution: the uniform bound from b is what terminates the argument.