Pointwise convergence of functions

Mathematical statement

This slice uses a complete finite domain and the symbolic tail f_n(x)=f(x)+c_x/(n+s). Supply an exact epsilon witness for each x. A maximum of the finitely many witnesses gives uniform convergence for this finite model, not for an arbitrary infinite domain. A finite prefix is insufficient.

Representative formula (exact TeX)

P(F)

Assumptions

• All f_n and f use the same explicitly specified finite domain and codomain.
• Epsilon is a positive rational and the tail certificate holds for every n≥N.

Symbol scope

• P — pointwise convergence
• F — function sequence

Intuition, definition, and misconceptions

N may depend on both x and epsilon.

A finite domain permits the finite maximum of all pointwise witnesses.

A finite prefix does not prove convergence, and the finite-domain implication does not extend automatically to infinite domains.

Do not exchange quantifiers to infer uniform convergence in general, or infer a limit from finitely many terms.

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

• Sequence — n indexes the functions.
• Function mapping — The domain and codomain are shared.
• Sequence limits and exact tail cutoffs — Each fixed point supplies a value sequence.
• Quantifiers and explicit domains — For every x and epsilon there is a witness N(x,epsilon).

Verification boundary

Only explicit finite domains and symbolic rational tails are handled.

not_formalized

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

Next obligation

Add uniform convergence and conditions for interchanging 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.