Suppose $(X,\mathcal O _X)$ is a complex manifold, $L$ a line budle with nontrivial global section $s\in H^0(X,L)$. Then in sheaf category, one has a inclusion $\mathcal O_X\rightarrow L$. Notice that it is equivalent to for every open set $U\in X$, $\mathcal O_X(U)\rightarrow L(U)$ is injective. So does it mean that if $0\neq s\in H^0(X,L)$, $s_{|U}\neq 0$?
I know for holomorphic functions, there exist identity theorem. But in my opinion, there exists some differences between the two notions (take open covering...) So, how should I prove the sheaf inclusion?
Thanks in advance.