Approximation by Continued Fractions
Melvyn B. Nathanson · Proceedings of the American Mathematical Society · 1974
Let $x$ be a real irrational number whose continued fraction has infinitely many partial quotients not less than $k$. A simple proof is given that the inequality $|x - p/q| < (k^2 + 4)^{-1/2} q^{-2}$ has infinitely many rational solutions $p/q$. The constant $(k^2 + 4)^{-1/2}$ is best possible.