= 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 sheaves>
$$
\mathcal O_X^r\longrightarrow\mathcal O_X,\qquad(h_i)\longmapsto\sum_i f_ih_i
$$
is <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 $1$ supplies
$$
\boxed{\sum_{i=1}^r g_if_i=1,\qquad g_i\in A.}
$$
This is the <unit-ideal certificate from a principal affine cover>.
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