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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | import OAI.Probability.InvariantIsing.Magnetic.MagneticGridProjection
namespace OAI
/-! The normalized radius of the magnetic grid projection converges to its mesh. -/
noncomputable section
open Filter
open scoped Topology
namespace InvariantIsing
lemma magnetic_cover_distance_tendsto (N : ℕ → ℕ) (hN : ∀ k, 0<N k)
(hNlim : Tendsto N atTop atTop) (A : ℕ) {δ : ℝ} (hδ : 0<δ) :
Tendsto (fun k => (⌈δ*N k+A⌉₊:ℝ)/(N k:ℝ)) atTop (𝓝 δ) := by
have he k : 0≤(⌈δ*N k+A⌉₊:ℝ)/(N k:ℝ)-δ ∧
(⌈δ*N k+A⌉₊:ℝ)/(N k:ℝ)-δ≤((A:ℝ)+1)/(N k:ℝ) := by
have hp : (0:ℝ)<N k := Nat.cast_pos.mpr (hN k)
have hx : 0≤δ*N k+A := by positivity
have hl := Nat.le_ceil (δ*N k+A)
have hu := Nat.ceil_lt_add_one hx
constructor
· apply sub_nonneg.mpr
apply (le_div_iff₀ hp).mpr
linarith [(Nat.cast_nonneg A : (0:ℝ)≤A)]
· have hdiff : (⌈δ*N k+A⌉₊:ℝ)/(N k:ℝ)-δ=
((⌈δ*N k+A⌉₊:ℝ)-δ*N k)/(N k:ℝ) := by field_simp
rw [hdiff]
exact div_le_div_of_nonneg_right (by linarith) hp.le
have hz := squeeze_zero (fun k => (he k).1) (fun k => (he k).2)
(tendsto_const_nhds.div_atTop (tendsto_natCast_atTop_atTop.comp hNlim))
simpa only [sub_add_cancel,zero_add] using hz.add_const δ
end InvariantIsing
end
end OAI
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