On differences of two harmonic numbers

We prove the existence of a set of positive real numbers X such that, for any x ∈ X, the difference of two harmonic numbers Hₘ - Hₙ can asymptotically never get too close to x: for some c=cₓ>0 and all sufficiently large m ≥ n ≥ N₀(x), one has the lower bound | (Hₘ - Hₙ) - x | ≥ c · n⁻². We show 1 ∈ X which answers a question of Erd{o}s and Graham. The set also contains the rational numbers 1/n ∈ X and 2/(2n+1) ∈ X.