# Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natural density of $X$ is zero?

It is a result of folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. Šalát's paper: Convergence of subseries of the harmonic series and asymptotic densities of sets of integers (1987), which in turn redirects to Šalát's On subseries for a different proof. However, I am convinced that the result is (much?) older than that, and was told by G. Grekos that Šalát himself in his talks presented it as well-known. So my question is:

Do you have any clue about the first (explicit) occurrence of the result in the literature?

• For the record, the question was first motivated by mathoverflow.net/questions/211177, but is hopefully interesting per se. Jul 9, 2015 at 19:34

I believe this is due to Kronecker. Namely, if you look at theorem 3 here, which is due to Kronecker, and says that if $$\sum_{n=1}^\infty a_n$$ is convergent, and $(p_n)_{n\geq 1}$ is an increasing and unbounded sequence, then $$\lim_{n\rightarrow \infty}\frac{p_1 a_1 + p_2 a_2 + \dotsc + p_n a_n}{p_n} = 0.$$ Now, let your set be $X =\{x_1, \dotsc, x_k, \dotsc\},$ in order. Set $a_n = 1/ x_n,$ while $p_n = x_n,$ your assertion follows.
• For the record, here is the full reference to the papers mentioned in the answer above: J. Krzyś, A theorem of Olivier and its generalizations (in Polish), Prace matem. 2 (1956), 159-164 and L. Moser, On the series $\sum 1/p$, Amer. Math. Monthly 65 (1958), 104-105. @Igor: G. Grekos noted there is a typo in the way the name of Krzyś is spelled in the post, and tried to edit, but I think he can't, since he is not a registered user. Jul 10, 2015 at 6:57