A theorem of Khintchine type

Let $ \psi : \mathbb{N} \to [0, \infty)$ be an approximation function with $ \sum_{q = 1}^ \infty q \psi (q)$ $ = \infty$ and the property that there exists $ \delta > 0$ such that $\psi(q) \geq \delta \psi(s)$ for all $q \in \mathbb{N}$ and all $s \in {q, q+1, \ldots, 2q\}$. Then the set $$\left\{x \in (0,1) : \left|x - \frac p q \right| < \psi(q) \textrm {forinfinitely many reduced rationals }\frac p q \right\}$$ has Lebesgue measure one.