I'm trying to prove this statement in Hartshorne's book:

**MY ATTEMPT**

**Injectivity part**

If $\varphi$ is an isomorphism, then $\varphi_U$ is an isomorphism of abelian groups for every open subset of $U$

So for $s\in F(U)$ and $t\in F(V)$, with $p\in U$ and $p\in V$, we have

$\varphi_p(s_p)=\varphi _p(t_p)\implies (\varphi_U(s))_p=(\varphi_V(t))_p$

where $\varphi_U(s)\in G(U)$ and $\varphi_V(t)\in G(V)$

Then there is an open subset $W\subset X$ such that $W\subset U\cap V$ and $\varphi_U(s)_{|W}=\varphi_V(t)_{|W}$.

I can't see why this implies $s_p=t_p$.

I need help.

Thanks a lot.