some questions in a paper by E. H. Neville (1949) about Farey series?

Question

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;

1. 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.

2. 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.

Answer 1

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.