I am learning computational geometry by myself using these lecture notes from ETH Zurich. Here is an exercise (8.9) I have been stuck for a few days:
For a line arrangement $A$ of a set of $n$ lines in $R^2$, let $F$ denote the union of the closure of all bounded cells. Show that the complexity (number of vertices and edges of the arrangement lying on the boundary) of $F$ is $O(n)$.
My intuition is to use induction, proving that when a new line $l$ is added, the number of added edges is a constant. This is similar to how we proved the zone theorem. However, I cannot figure it out. Can someone provide me with some hints?