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It is a result from additive-theory 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 the result as well-known. So my question is:

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

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?

It is a result from additive-theory 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 the result as well-known. So my question is:

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

It is a result from additive-theory 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 the result as well-known. So my question is:

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