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Erdős problem 480

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Let x1,x2,…∈[0,1]x_1,x_2,\ldots\in [0,1] be an infinite sequence. Is it true that inf⁡nlim inf⁡m→∞n∣xm+n−xm∣≤5−1/2≈0.447?\inf_n \liminf_{m\to \infty} n \lvert x_{m+n}-x_m\rvert\leq 5^{-1/2}\approx 0.447? A conjecture of Newman.

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480.lean

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Let x1,x2,…∈[0,1]x_1,x_2,\ldots\in [0,1] be an infinite sequence. Is it true that inf⁡nlim inf⁡m→∞n∣xm+n−xm∣≤5−1/2≈0.447?\inf_n \liminf_{m\to \infty} n \lvert x_{m+n}-x_m\rvert\leq 5^{-1/2}\approx 0.447? A conjecture of Newman.

FormalConjectures/ErdosProblems/480.leanErdos480.erdos_4803 linesExact file
True ↔  ∀ (x : ℕ → ℝ),    (∀ (n : ℕ), x n ∈ Set.Icc 0 1) → ⨅ n, Filter.liminf (fun m => ↑↑n * |x (m + ↑n) - x m|) Filter.atTop ≤ 1 / √5
SolvedStatement only, no proof