import Mathlib

namespace OAI


noncomputable section
open Set Filter
open scoped Topology

namespace WeakMTWTransport

lemma diagonal_tendsto_of_close {X : Type*} [PseudoMetricSpace X]
    {f a : ℕ → X} {x : X} {ε : ℕ → ℝ}
    (ha : Tendsto a atTop (𝓝 x)) (hε : Tendsto ε atTop (𝓝 0))
    (hclose : ∀ᶠ k in atTop,dist (f k) (a k)≤ε k) : Tendsto f atTop (𝓝 x) := by
  apply tendsto_of_tendsto_of_dist ha
  exact squeeze_zero' (Eventually.of_forall (fun _=>dist_nonneg)) (hclose.mono (fun k hk=>by simpa only [dist_comm] using hk)) hε

end WeakMTWTransport

end

end OAI