Remarks on a Problem of Obreanu

Let a 1 < a 2 <. .. be any sequence of integers. Assume that the infinite sequence of condition : To every E > for all n > n and all k o (1) ~u n+-u n I < E a k Obreanu asked (Problem P. 35 Can. Math. Bull .) under what conditions on the sequence a < a 2 <. .. does (1) imply that 1 is convergent. N. G. de Bruijn and P. Erdős the sequence u n proved that a necessary and sufficient the convergence of u is that the sequence n and that the greatest common divisor of the condition for (1) to (2) lim lim I u n+a-u n I= 0 n-00 r r numbers u satisfies the following n 0 there is an n = n () such that 0 0 267 { a } be infinite n a should be 1I n imply The condition (1) is very strong and is "nearly equivalent" to Cauchy' s criterion for convergence. We discuss various conditions which are weaker than (1) .