TITLE: 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?
QUESTION [10 upvotes]: 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?

REPLY [11 votes]: 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. 
I should note that in (one of) his papers, Salat attributes the result independently to Leo Moser (Monthly, 1958, DOI: 10.2307/2308884), and Krzyś (Prace Matem 1956) - I could not find the latter paper. Neither can I find the original Kronecker paper.