Regarding property of Farey sequence

Question

I am studying apostol Dirichlet series and modular functions book and struck upon this question. Question is - Two reduced fractions $a/ b$ and $c/d$ are called similarly ordered if $(c-a)(d-b) \ge 0$. Prove that two neighbors $a_i / b_i$ and $a_{i+1}/ b_{i+1}$ in Farey fractions are similarly ordered. I tried using property $bc-ad=1$ for consecutive fractions but could prove completely.

Answer 1

If $c<a$ and $d>b>0$, then $$\frac cd<\frac ad<\frac ab,$$ contraditcion.

If $c>a$ and $0<d<b$, then $$1=bc-ad>bc-cd=c(b-d)\ge 1\cdot 1, $$ contradiction.