 Solution

Taking determinants of the convergent matrices gives $q_n{p}_{n-1}-q_{n-1}{p}_n=(-1{)}^n$. The complete-quotient formula then yields $|\alpha -p_n/q_n|<1/(q_n{q}_{n+1})<1/q_n^2$. 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 $1/\sqrt{5}$ bound).