Solution
= Solution
Taking <determinants> of the convergent <matrices> gives $q_np_{n-1}-q_{n-1}p_n=(-1)^n$. The complete-quotient formula then yields $|\alpha-p_n/q_n|<1/(q_nq_{n+1})<1/q_n^2$. Comparing two consecutive errors shows at least one is below $1/(2q^2)$; hence both stated inequalities occur infinitely often (the second is the elementary precursor of Hurwitz's sharper $1/\sqrt5$ bound).