Basic definition of Inverse Limit in sheaf theory / schemes I read the book "Algebraic Geometry" by U. Görtz and whenever limits are involved I struggle for an understanding. The application of limits is mostly very basic, though; but I'm new to the concept of limits.
My example (page 60 in the book): Let $A$ be an integral domain. The structure sheaf $O_X$ on $X = \text{Spec}A$ is given by $O_X(D(f)) = A_f$ ($f\in A$) and for any $U\subseteq X$ by 
\begin{align}
O_X(U) &= \varprojlim_{D(f)\subseteq U} O_X(D(f)) \\ 
&:= \{ (s_{D(f)})_{D(f)\subseteq U} \in \prod_{D(f)\subseteq U} O_X(D(f)) \mid  \text{for all } D(g) \subseteq D(f) \subseteq U: s_{D(f)\big|D(g)} = s_{D(g)}\} \\ 
&= \bigcap_{D(f)\subseteq U} A_f.
\end{align}
I simply don't understand the last equality: In my naive understanding the elements of the last set are "fractions" and the elements of the Inverse Limit are "families of fractions".
Any hint is appreciated.

Is not the best way of defining the structure sheaf. It just uses the fact that $D(f)$ form base for the topology. To see the last isomorphism if you want you can define the sheaf of regular functions $\mathscr{R}(U)=\cap_{x\in X}A_{\frak{p}_x}$. The correct way of understanding this is that for each open it gives you the ring of regular functions on $U$ and that can be seen it as functions that are regular at each point of $U$. Notice that $\mathscr{R}(U)$ is the same as the last intersection and you have a natural map $\mathscr{R}\rightarrow \mathscr{O}_X$ that for $U\subset X$ sends $\phi\in \cap_{D(f)\subset U}A_f$ to $(\phi\mid D(f)\subset U)\in \prod A_f$ that stalk-wise is an isomorphism. 
