Uniform convergence of functions

Mathematical statement

Use a finite domain and symbolic rational tails. The maximum of all pointwise witnesses gives a uniform witness on this complete finite domain, not on arbitrary infinite domains. Check exact supremum gaps and the all-n tail; a finite prefix is insufficient.

Representative formula (exact TeX)

U(F)

Assumptions

• The finite domain and codomain are shared, and epsilon is a positive rational.
• One N controls every point and every n≥N.

Symbol scope

• U — uniform convergence
• F — function sequence

Intuition, definition, and misconceptions

N must not depend on x.

A finite domain permits the maximum of pointwise witnesses.

A finite prefix does not prove the tail, and the finite-domain argument does not extend automatically to an infinite domain.

Separate pointwise witnesses are not a common witness on an arbitrary domain, and finitely tested points cannot certify an infinite-domain supremum.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Pointwise convergence of functions — Finite-domain pointwise witnesses supply the maximum construction.
• Metric space — Errors are measured in the specified metric.
• Absolute value — Real-valued supremum errors use absolute values.
• Quantifiers and explicit domains — Uniformity fixes the quantifier order.

Verification boundary

The universal certificate is limited to the explicit finite-domain symbolic tail.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add continuity of uniform limits.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.