 Solution

If all but finitely many convergents had $b_n>1/\sqrt{5}$, then eventually $c_n>1/\sqrt{5}$. Part (b) would imply $c_n{a}_{n+1}\le \sqrt{1-4c_n^2}<c_n$, impossible since $a_{n+1}\ge 1$. Thus infinitely many $b_n\le 1/\sqrt{5}$, which is exactly $|\theta -p_n/q_n|\le 1/(\sqrt{5}{q}_n^2)$.