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

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At a finite maximal time an ODE solution leaves every compact subset of the domain

Statement

Let $x : (α, β) \to R^{n}$ be the maximal solution of an IVP whose continuous vector field is locally Lipschitz in the state variable on its open ODE domain. If the positive maximal endpoint is finite, the solution must eventually leave every compact set $K$ contained in that domain: some $s < β$ satisfies $(t, x (t)) \notin K$ for every $s < t < β$ . The analogous assertion holds as $t ↓ α$ when $α$ is finite.

At a finite maximal endpoint the solution leaves every compact subset of the ODE domain. If the positive maximal endpoint is finite, the solution must eventually leave every compact set.

Facts & Assumptions

Given: A Picard–Lindelöf maximal solution with a finite endpoint and an arbitrary compact subset $K$ of the open ODE domain.

[L1]

A solution of a Picard–Lindelöf ODE whose graph has a sequence approaching a compact interior endpoint state extends past that endpoint (A solution whose graph approaches a compact interior region at a finite endpoint extends past that endpoint).

[L2]

Countable choice selects one member from each nonempty set in a family indexed by $N$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[L3]

Every nonempty subset of $N$ has a least element (The well-ordering principle).

[L4]

Recursion on $N$ defines a unique sequence from a specified initial value and successor rule (The recursion theorem).

Proof

technique · contradiction

1.1

Suppose, for contradiction, that the graph does not eventually leave $K$ . For each positive integer $j$ , the set of $t \in (β - 1 / j, β)$ with $(t, x (t)) \in K$ is nonempty, so [L2] selects one $t_{j}$ from each set. Then $t_{j} \to β$ from below. Starting with index one, [L3] gives the least later index whose term exceeds the preceding selected term, and [L4] recursively defines these indices; the resulting increasing subsequence tends to $β$ .

givenL2L3L4assume-contrachoose

2.1

By [L1] the solution extends past $β$ , contradicting maximality; reflecting time gives the finite-left-endpoint assertion.

step 1.1L1discharge-contradiction∎

Depends on

Used by

Dependency tree · two levels

25 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)