I have started reading scheme theory form "The Geometry of Schemes" written by Eisenbud and Harris.
Let $\mathscr{F}_x=\lim\limits_{\longrightarrow \\ x\in U}\mathscr(U)$, which is the disjoint union of all open sets $U$ containing $x$, with this equivalence relation: $\sigma\sim\tau$, if $\sigma \in \mathscr{F}(U)$ and $\tau\in \mathscr{F}(V)$ and there exists an open neighborhood of $x$, such as $W\subset U\cap V$ so that $res_{U,W}(\sigma)=res_{V,W}(\tau)$. (Where $res_{U,W}$ and $res_{V,w}$ are restriction maps.)
My problem is that I don't understand the difference between $\bigcup\mathscr{F}_x$ and $\mathscr{F}(U)$. My attempt: Let $\rho$ and $\sigma$ be two different global section of $\mathscr{F}(U)$. If $\rho$ and $\sigma$ represent the same element in $\bigcup\mathscr{F}_x$, it means that for all members of $U$, such as $y$, we have $\rho_y=\sigma_y$. For all $y\in U$ we define $U_y$ to be the open neighborhood of $y$, where $\rho|_{U_y}=\sigma|_{U_y}$. These $U_y$s form an open covering for $U$. If from the previous sentence we can conclude that $\rho$ and $\sigma$ are the same, then they would represent the same member in $\mathscr{F}(U)$ as well. However I'm not sure if it's true that if two sections of a sheaf agree on an open covering, then we can say they are the same in $\mathscr{F}(U)$