I am reading the proof of
Let $\mathcal M=\widetilde M$ be a quasicoherent sheaf on Spec$A$. Then $\mathcal M$ is of finite type if and only if $M$ is finitely generated.
And I get stuck at the line
Let $A$ be a ring, $M$ be an $A$-module, $A^n\to M$ be a morphism. Let $\cup_{i=1}^n D(f_i)$ be an open cover of spec$A$. Then $A^n\to M$ is surjective if $A^n[f_i^{-1}]\to M[f_i^{-1}]$ is surjective.
I cannot figure out why. In fact I know surjectivity is local property, i.e. the map between two $A$-modules is surjective if and only if it is surjective when localizing at every prime ideal of $A$. So I try to use $M_p=lim_{f_i \notin p}M[f_i^{-1}]$. If the inverse limit preserves surjectivity then it seems finishes the proof. Is this right? Or if there is any other approach?