The main result of the paper is the following generalization of G. E. H. Reuter’s lemma [Acta Math. 97, 1–46 (1957; Zbl 0079.34703)]. Let be a sequence of real numbers satisfying and
where and are all nonnegative. Then is bounded if and only if where is defined recursively by and for with
As an application the authors give an alternative proof of a special case of Theorem 1.1 of [M. Chen, Chin. Ann. Math., Ser. B 20, No. 1, 77–82 (1999; Zbl 0922.60068)] concerning upwardly skip-free processes. The authors use their generalization of Reuter’s lemma and obtain some new results for downwardly skip-free chains, such as Markov branching processes. Finally, they study asymptotic birth-death processes being neither upwardly nor downwardly skip-free.