Does every set $A\subseteq \mathbb{N}$ of positive density contain some finite $S\subset A$ such that ${\sum }_{n\in S}\frac{1}{n}=1$?

AMS 11 40 · solved (literature)

theorem erdos_299 : answer(False) ↔ (∃ (a : ℕ → ℕ),
    StrictMono a ∧ (∀ n, 0 < a n) ∧
    (fun n ↦ (a (n + 1) : ℝ) - a n) =O[atTop] (1 : ℕ → ℝ) ∧
    ∀ S : Finset ℕ, ∑ i ∈ S, (1 : ℝ) / a i ≠ 1)