# Covering scheme by affines, $X = \bigcup X_f$

I am reading lemma 27.27.3 in stacks project.

In the proof it seems it seems to claim:

If $$X$$ is a scheme, where $$f_1,\ldots, f_n \in \Gamma(X, O_X)$$ generates the ring. Then $$X= \bigcup X_{f_i}$$ where $$X_f:= \{x \in X \, : \, f_x \not= 0 \in O_{X,x} \}$$

How so? Suppose false, then pick $$x$$ not in the union. Then $$1_x =\sum (g_if_i)_x =0 \in O_{X,x}$$ But there is nothing wrong this either...