Is a locally finitely generated sheaf of modules finitely generated on sections? Everywhere I have looked, I have seen the following definition of a finitely generated sheaf of modules.
"Let $(X, \mathcal{O})$ be a ringed space. A sheaf of modules $\mathcal{F}$ on $X$ is finitely generated if for all $a \in X$ there exists a neighbourhood $U$ of $a$, an integer $n$ and a surjective morphism $\phi: \mathcal{O}_U^n \to \mathcal{F}_U$."
Here $\mathcal{O}_U$ and $\mathcal{F}_U$ denote the restricted sheaves.
One would like to conclude the following: for any point $a \in X$ there exists a neighbourhood $U$ of $a$ such that for all $V \subseteq U$ the morphism on sections $\phi(V): \mathcal{O}^n(V) \to \mathcal{F}(V)$ is surjective. This is the only conclusion I could draw so that the notion of "finitely generated" has some intuitive meaning. However, it is well known that a surjective morphism need not be surjective on sections (I have seen a few examples here on stackexchange).
My question is: is this claim true and, if yes, how would one go about proving it? Thanks a lot.

This won't be true in general, although in reasonable situations $a$ will have a neighbourhood basis of $V$ such that $\mathscr O^n(V) \to \mathscr F(V)$ is surjective.  In the context of varieties or schemes, one should assume that $\mathscr F$ is coherent; then if $V$ is an affine n.h. of $a$, this surjectivity will hold (the point being that the kernel $\mathscr K$ will again be coherent, and $H^1$ of $\mathscr K$ will vanish on an affine $V$).
In the context of complex analytic spaces, one has a similar (but deeper) fact, taking $V$ to run through a n.h. basis consisting of Stein spaces.
