Erdős Problem 488

Reference: erdosproblems.com/488

theorem Erdos488.erdos_488 : (∀ (A : Finset ℕ), A.Nonempty → 0 ∉ A → 1 ∉ A → ∀ (n m : ℕ), m > n → A.max ≤ ↑n → ↑{x ∈ Finset.Icc 1 m | x ∈ {n : ℕ | n ≥ 1 ∧ ∀ a ∈ A, ¬a ∣ n}}.card / ↑m < 2 * ↑{x ∈ Finset.Icc 1 n | x ∈ {n : ℕ | n ≥ 1 ∧ ∀ a ∈ A, ¬a ∣ n}}.card / ↑n) ↔ False

Let $A$ be a finite set and $$B = \{n \ge 1 : a \nmid n \text{ for all } a \in A\}.$$ Is it true that, for every $m > n \ge \max(A)$, $$\frac{|B \cap [1, m]|}{m} < 2 \frac{|B \cap [1, n]|}{n}?$$