Separable subgroups have bounded packing

In this note, we prove that separable subgroups have bounded packing in ambient groups. The notion bounded packing was introduced by Hruska and Wise and in particular, our result answers positively a question of theirs, asking whether each subgroup of a virtually polycyclic group has the bounded packing property.
Comment: 2 pages, to appear Proc. Amer. Math. Soc