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.

My personal advice is to study a bit of category theory: it will let you understand all this stuff in a very clearer way. In fact you can easily realize that the first equality is not a definition, but a way to express a limit of an arbitrary presheaf, while the second is an isomorphism, not exactly an equality, given by the universal property defining limits. I started with Hartshorne, but without category theory as a background it's just like wandering in the dark without even a candle with you.
