LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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Euler polygonal approximations for a continuous ODE are uniformly bounded and equicontinuous

Statement

Let $h > 0$ , and let $F$ be continuous on a compact cylinder $[t_{0}, t_{0} + h] \times \overline{B} (x_{0}, r)$ and bounded there by $M$ , with $h M \leq r$ . The Euler polygonal approximations formed with positive mesh sizes tending to zero remain in the cylinder, are uniformly bounded, and are equicontinuous. More precisely, every approximation is $M$ -Lipschitz. The final mesh cell may be shorter than the others.

Euler polygonal approximations on a compact cylinder are uniformly bounded and equicontinuous.

Facts & Assumptions

Given: The compact cylinder, its vector-field bound, and the Euler recursion at mesh vertices.

[L1]

A family of $R^{n}$ -valued curves is equicontinuous when, for every $ε > 0$ , one $δ > 0$ makes $∥ x_{m} (t) - x_{m} (s) ∥_{2} < ε$ for every member whenever $∣ t - s ∣ < δ$ .

Proof

technique · direct

1.1

Recursively set the next Euler vertex using the field at the preceding vertex and interpolate linearly. Induction gives displacement at most $M$ times elapsed time, and $h M \leq r$ keeps every vertex and every interpolated segment inside the cylinder, including a shortened final cell.

givenalgebra

2.1

Each linear segment has slope norm at most $M$ , and summing across intervening mesh cells gives $∥ x_{m} (t) - x_{m} (s) ∥_{2} \leq M ∣ t - s ∣$ ; this common estimate gives uniform boundedness and [L1], with constant polygons when $M = 0$ .

step 1.1L1algebra∎

Depends on

First-order systems, initial value problems, and solutions on intervals

Used by

Peano local existence for a continuous first-order system Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, Ch. 2 (standard reference, not scraped)
Jiri Lebl, Basic Analysis I, Section 6.3 (standard reference, not scraped)