If an irrational number $\theta$ lies between two consecutive terms $a/b$ and $c/d$ of the Farey sequence of order n, prove that at least one of the following holds: $|\theta- a/b| < 1/2b^2$ or $|\theta - c/d| < 1/2 d^2$.
How can we about proving such a statement?
Hint: $\frac cd-\frac ab=\frac1{bd}\le\frac{1/b^2+1/d^2}2$ by the arithmetic-geometrix-mean inequality.