After a few discussions regarding lattices, fundamental lattices and some of their properties, the author writes this proof of a theorem regarding Farey series. I don't really see how some of his claims follow. They're not so obvious to me. I get how only visible points inside or on the boundary represent all the fractions of the Farey series, but how does he know that there won't be any representative point inside the triangle OPP' ? Also, what about the case where the two consecutive fractions are $\frac{0}{1}$ and $\frac{1}{n}$ ? We don't have a triangle in that case.