= Solution
Consecutive convergents obey the standard error estimate
$$
0<\left|\alpha-\frac{p_n}{q_n}\right|
<\frac1{q_nq_{n+1}}
\leq\frac1{a_{n+1}q_n^2}.
$$
If the partial quotients are unbounded, then for every $c>0$ there are infinitely many indices with $a_{n+1}>1/c$; otherwise the whole sequence would be bounded after adjoining its finite initial segment. At each such index,
$$
\left|\alpha-\frac{p_n}{q_n}\right|<\frac c{q_n^2}.
$$
The denominators of the infinite continued fraction tend to infinity, so these give infinitely many distinct rational numbers of the required kind.
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