TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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Locally uniform limits of harmonic functions are harmonic

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ .

If harmonic $u_{j}$ converge locally uniformly on $Ω$ to $u$ , then $u$ is harmonic.

Proof

Given: Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) and $u_{j} \to u$ uniformly on compact subsets of $Ω$ .

1.1

On every $B_{r} (x) ⋐ Ω$ , Ball mean-value property for harmonic functions under Countable Choice says $u_{j} (x) = A_{u_{j}} (x, r)$ under Countable Choice [given].

1.2

Uniform convergence on $\overline{B_{r} (x)}$ permits passage to the integral, giving $u (x) = A_{u} (x, r)$ [given].

2.1

The locally uniform limit $u$ is continuous. Continuous ball-mean-value functions are harmonic therefore applies under the same Countable Choice hypothesis and proves $u$ harmonic [step 1.2]. ∎

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