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.
