A variation on arithmetic continuity

A sequence (x(k)) of points R, the set of real numbers, is called arithmetically convergent if for each epsilon > 0 there is an lat for every integer m, we have vertical bar x(m) - x(<m,n>)vertical bar < epsilon, where k vertical bar n means that k divides n or n is a multiple of k, and the symbol < m, n > denotes the greatest common divisor of the integers m and n. We prove that a subset of R is bounded if and only if it is arithmetically compact, where a subset E of R is arithmetically compact if any sequence of point in E has an arithmetically convergent subsequence. It turns out that the set of arithmetically continuous functions on an arithmetically compact subset of R coincides with the set of uniformly continuous functions where a function f defined on a subset E of lit is arithmetically continuous if it preserves arithmetically convergent sequences, i.e., (f (x(n)) is arithmetically convergent whenever (x(n)) is an arithmetic convergent sequence of points in E.