Solution
= Solution
If all but finitely many convergents had $b_n>1/\sqrt5$, then eventually $c_n>1/\sqrt5$. Part (b) would imply $c_na_{n+1}\le\sqrt{1-4c_n^2}<c_n$, impossible since $a_{n+1}\ge1$. Thus infinitely many $b_n\le1/\sqrt5$, which is exactly $|\theta-p_n/q_n|\le1/(\sqrt5q_n^2)$.