Double dual of a subsheaf of a vector bundle

Question

Let $F$ be a non-trivial proper subsheaf of a vector bundle $E$ over a smooth projective surface $X$. Is it necessarily true that $F^{**} \subset E$?

I can construct a map from $F^{**} \to E$ from dualizing twice the sequence $0 \to F \to E \to E/F \to 0$.

But why in the left we should end up with $0$?

Answer 1

At the general point of $X$ one has $F^{**} = F$, hence the map $F^{**} \to E$ is generically injective, hence its kernel is a torsion sheaf. But $F^{**}$ is torsion free, hence the map $F^{**} \to E$ is injective.