Unbounded continued-fraction coefficients exclude quadratic irrationality

For an irrational $x$ with continued fraction convergents $p_n/q_n$, one has $|x-p_n/q_n|\le 1/(q_n{q}_{n+1})\le 1/(a_{n+1}{q}_n^2)$. The Liouville approximation theorem for a quadratic irrational gives a positive lower bound $C/q_n^2$ instead. Thus unbounded partial quotients preclude quadratic irrationality, and in particular imply $x^2$ is irrational.