Good points for diophantine approximation

Given a sequence (x_n)^∞_n=1 of real numbers in the interval [0,1) and a sequence (δ_n)^∞_n=1 of positive numbers tending to zero, we consider the size of the set of numbers in [0,1] which can be 'well approximated' by terms of the first sequence, namely, those y ∈ [0,1] for which the inequality |y - x_n| < δ_n holds for infinitely many positive integers n. We show that the set of 'well approximable' points by a sequence (x_n)^∞_n=1, which is dense in [0,1], is 'quite large' no matter how fast the sequence (δ_n)^∞_n=1 converges to zero. On the other hand, for any sequence of positive numbers (δ_n)^∞_n=1 tending to zero, there is a well distributed sequence (x_n)^∞_n=1 in the interval [0,1] such that the set of 'well approximable' points y is 'quite small'.