I don't understand the following criterion for an open cover related to a subset of the coordinate ring. Suppose $S\subset k[V]$ is a subset of the coordinate ring of a closed set $V$.
Why is it that $V\subseteq\bigcup_{f\in S} Z_f\iff \langle S\rangle=k[V]$? Here $Z_f$ is the principal open set, and I'm writing $\langle S\rangle$ to denote the ideal generated by $S$ in $k[V]$.
There a lot of ways to see this. You could take the left-hand side and just start expanding: turn each $Z_f$ into a complement, use De Morgan, ...
But I want to say in plain English why this is a natural result. $V = \bigcup Z_f$ means that at each point of $V$ there's some $f$ that doesn't vanish. Another way to say this is that the functions in $S$ have no common zeros, and the Nullstellensatz says that — up to taking radicals — ideals are determined by their zero sets.
So hopefully after toying with the given union and your statement of the Nullstellensatz you can write down a proof.