Hey math experts, I need a little help with a proof.
I found a sketch of the proof I want to do on the web(attached) , and I find it difficult to understand.

VC dimension and epsilon-nets-week-cut-lemma.jpg


1. The definition of E. (I spouse there is a mistake there and it should be E:={{ l \in L : l \cap e \not= \emptyset } : e is segment}) what do they mean by "e is a segment"?

2. What is the explanation to the existence of such a " 2/3r-net" of \Sigma?

3. Why does the fact that R is a 2/3r-net means that each triangle does not intersect more than n/r lines?

Any help will be appreciated.