 Solution

The quasi-compactness of $X$ gives a finite cover by the affine principal neighbourhoods constructed above. Because at every point some $f_i$ is a unit in the local ring, the map of coherent sheavesis surjective. Its kernel $\mathcal{F}$ is a coherent submodule of the trivial bundle, so $H^1(X,\mathcal{F})=0$ by part (a). The long exact sequence in sheaf cohomology makes the map on global sections surjective. Lifting $1$ suppliesThis is the unit-ideal certificate from a principal affine cover.