Is it true that if $A\subset \mathbb{N}\backslash \{1\}$ is a finite set with ${\sum }_{n\in A}\frac{1}{n}<2$ then there is a partition $A=A_1\bigsqcup A_2$ such that$$\sum _{n\in A_i}\frac{1}{n}<1$$for $i=1,2$?

AMS 5 11 · solved (literature)

theorem erdos_316 : answer(False) ↔ ∀ A : Finset ℕ, 0 ∉ A → 1 ∉ A →
    ∑ n ∈ A, (1 / n : ℚ) < 2 → ∃ (A₁ A₂ : Finset ℕ),
      Disjoint A₁ A₂ ∧ A = A₁ ∪ A₂ ∧
      ∑ n ∈ A₁, (1 / n : ℚ) < 1 ∧ ∑ n ∈ A₂, (1 / n : ℚ) <  1