  In 1974, M. B. Nathanson proved that every irrational number $\alpha$
represented by a simple continued fraction with infinitely many elements
greater than or equal to $k$ is approximable by an infinite number of rational
numbers $p/q$ satisfying $|\alpha-p/q|<1/(\sqrt{k^2+4}q^2)$. In this paper we
refine this result.
