Convex Lemma-lemma-2.jpg

I'm having trouble understanding this Lemma.
How can it be that taking a point to P, still be inside the union of A_i.
I Also attached a picture that shows what can be a possible example for the theorem.
I must be thinking Something wrong because in my example the segment ]p,y[ is not inside A_i.
I appreciate any help.

Convex Lemma-lemma-2-sce.jpg