Let p_n/q_n=(p_n/q_n)(x) denote the n^th simple continued fraction convergent to an arbitrary irrational number x∈ (0,1). Define the sequence of approximation constants θ_n(x):=q_n²|x-p_n/q_n|. It was conjectured by Lenstra that for almost all x∈(0,1), lim_n➜∞(1/n)|{j:1≦ j≦ n and θ_j(x)≦ z}|=F(z) where F(z) := z/log 2 if 0≦ z≦ 1/2, and (1/log 2)(1-z+log(2z)) if 1/2≦ z≦ 1. This was proved in [BJW83] and extended in [Nai98] to the same conclusion for θ_kj(x) where k_j is a sequence of positive integers satisfying a certain technical condition related to ergodic theory. Our main result is that this condition can be dispensed with; we only need that k_j be strictly increasing.