Convergence from Squared Extended-Norm Convergence

Proved

LevyStochCalc.Brownian.Ito.tendsto_of_tendsto_sq_enorm_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concavevendored-levystochcalc

Let E be a normed additive commutative group, let u : ℕ → E, and let w : E. If

\left(\|u_i-w\|_+\!:\!\mathbb{R}_{\ge0\infty}\right)^2\to0

along the natural numbers, then u_i → w in E. The declaration tendsto_of_tendsto_sq_enorm_readable converts convergence of squared extended norms into ordinary topological convergence.

Original Lean declaration header

Within namespace LevyStochCalc.Brownian.Ito. Shown in its original module context. Proof omitted.

theorem tendsto_of_tendsto_sq_enorm {E : Type*} [NormedAddCommGroup E] {u : ℕ → E} {w : E}
    (h : Filter.Tendsto (fun i => (‖u i - w‖₊ : ℝ≥0∞) ^ 2) Filter.atTop (𝓝 0)) :
    Filter.Tendsto u Filter.atTop (𝓝 w)

Module navigation

Module: LevyStochCalc/Ito/ItoFormulaUnboundedClamp.

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Source excerpt, lines 339–341 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Ito_ItoFormulaUnboundedClamp

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Ito/ItoFormulaUnboundedClamp.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open scoped NNReal ENNReal Topology
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  {n d : ℕ} {ℱ : MeasureTheory.Filtration ℝ ‹MeasurableSpace Ω›}

Formal statement


theorem LevyStochCalc.Brownian.Ito.tendsto_of_tendsto_sq_enorm_readable {E : Type*} [NormedAddCommGroup E] {u : ℕ → E} {w : E}
    (h : Filter.Tendsto (fun i => (‖u i - w‖₊ : ℝ≥0∞) ^ 2) Filter.atTop (𝓝 0)) :
    Filter.Tendsto u Filter.atTop (𝓝 w) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Ito/ItoFormulaUnboundedClamp.lean#L338-L354 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Ito.ItoFormulaUnboundedClamp`, declaration `LevyStochCalc.Brownian.Ito.tendsto_of_tendsto_sq_enorm`; the repository is private at upload time). Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (Apache-2.0; upstream copyright notice retained in the Lean text).

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