- Can you point me out the rigorous definition of "limit of a sequence of arcs" on a surface?
- In which topological space are we considering the limit of arcs to obtain a lamination?
Suppose that $S$ is a orientable surface of finite topological type, equipped with a hyperbolic metric of finite volume. Consider the set $A$ of simple geodesic arcs properly embedded in $S$. Consider also the larger set $C$ of closed subsets of $S$. We equip $C$ with the Hausdorff topology, to make it a metric space.
Suppose that $a_n \in A$ is a sequence of arcs. This also gives a sequence in $C$. If $a_n$ has a limit in $C$, then that is the desired object. It is a theorem that the limit is geodesic lamination: a closed subset of $S$ that can be realized as a disjoint union of simple geodesics.
It is an easy exercise to find a geodesic lamination that is not a Hausdorff limit of arcs.
- Where can I find a proof of the statement "every sequence of arcs converges to a lamination on a hyperbolic surface"?
Surely this is discussed in the reference you are reading. If you give a link to it, perhaps we can find the location of this result.
This is also discussed in Thurston's notes.