some questions in a paper by E. H. Neville (1949) about Farey series? I am reading the paper 
MR0029924: Neville, E. H. The structure of Farey series. Proc. London Math. Soc. (2) 51, (1949). 132–144. (Reviewer: W. H. Simons)
and by now two questions raised for me; 


*

*for the proof of Theorem 1, it says that:


"This theorem is an immediate corollary of the pair of inequalities..."
but it takes almost one page for me to prove it. Is there more simple way that I missed it.


*It seems Theorem 3 instead of


"...difference $|vx—uy|$ has the value 1 if and only if ..."
must be
"...difference $|vx—uy|$ has the value greater than 1 if and only if ..."
Is there any comments? thanks.



For Theorem 1, from $bu-av\ge 1$ and $cv-du\ge 1$ we have $$cbu-cav\ge c$$ and $$acv-adu\ge a.$$ Adding these eliminates $v,$ giving $u=u(bc-ad)\ge c+a.$
Similarly by eliminating $u$ we obtain $v=v(-ad+bc)=d(bu-av)+b(cv-du)\ge d+b.$
Addendum (after acceptance). For Theorem 3, if $|vx-uy|=1$ then by Theorem 1, $no$ rational  between $u/v$ and $x/y$ can have a denominator less than $v+y.$ You are right.
