Nathanson gives a proof in Elementary Methods in Number Theory (Theorem 7.14) that, if a set $S$ of positive integers has $$ \sum_{s\in S}\frac1s<+\infty $$ then the set of positive multiples of $S$ has a natural density, i.e., $$ \lim_{x\to+\infty}\frac{|\{m\in M(S):\ m\le x\}|}{x} $$ exists. I'd like to cite this result; does anyone know when it was first stated or proved? ("As far as I know Nathanson was first" would be a valid answer.)
Edit: Hall (Sets of Multiples) states that this is the best possible result in the sense that for any $\xi(x)\to+\infty$ there are sets $S$ with $$ \sum_{s\in S,s\le x}\frac1s<\xi(x) $$ for which the set of multiples of $S$ does not have a density. But I'm still looking for a reference on the original.