In the book Algebraic Geometry and Arithmetic Curves, Qing Liu state the following proposition(page 36) :
Proposition 2.12. Let $\alpha: \mathcal{F}\rightarrow \mathcal{G}$ be a morphism of sheaves on X. Then $\alpha$ is an isomorphism if and only if $\alpha_{x}$ is an isomorphism an isomorphism for every $x\in X$.
In order to show that $\alpha(U)$ is injective for every $U$, he take $t\in \mathcal{G}(U)$ and claim :
Then we can find a covering of $U$ by open sets $U_i$ and sections $s_i \in \mathcal{F}(U_i)$ such that $\alpha(U_i)(s_{i})=t|_{U_i}$
My question is : How do we know that there exists such covering $U_i$ ?
For every $x \in U$ we have that the germ $t_x \in G_x$ lies in the image of $\alpha_x$, i.e. there is some open neighborhood $U_x$ of $x$ and some section $s \in F(U_x)$ with $\alpha_x(s_x)=t_x$. After shrinking $U_x$, we even have $\alpha(s)=t|_{U_x}$.