Talk:Uniform limit theorem

The proof given is not valid. It is only valid in a metric space. If the theorem as stated is true, the proof should hold in any space where a uniform limit is meaningful, namely a uniform space. — Preceding unsigned comment added by 69.165.203.213 (talk) 05:49, 30 May 2017 (UTC)

Uniform limit theorem also holds for *locally-uniformly convergent sequences* without weakening the theorem

Since continuity is a local property, global uniform convergence can be replaced with local uniform convergence. Example: $\lim _{n\to \infty }e^{x-n}=0$ is continuous, but the sequence is not globally uniformly convergent.

This should be obvious (proof is almost exactly the same), but unfortunately the best source I can find for it is this question on stack exchange without any meaningful traffic.

Using , the locally uniform convergence of (locally) uniform functions converges to a (locally) uniform function. Tokarak (talk) 18:02, 21 March 2026 (UTC)