Let an be a non-increasing real sequence such that converges; then clearly an ↓ 0. We shall ignore the trivial case where an = 0 for all large n, and so we assume that an > 0 for all n, from now onwards. In  J. B. Wilker introduced certain new sequences associated with the rate of convergence of , and obtained various relations between them, in order to investigate packing problems in convex geometry. Let us define
We write P, Q and T respectively for the inferior limits of pn, qn and tn, and P1, Q1 and T1 for the corresponding superior limits. Further, we put
It is immediately clear from these definitions and our assumptions about an that
the latter since nan → 0 by Olivier's theorem [2, p. 124].