Sequence limits and exact tail cutoffs

Mathematical statement

A real sequence u:N0→R tends to L when, for every ε>0, there is a natural N≥0 such that |u(n)−L|<ε for all n≥N. Here u(n)=L+c/(n+d), with d≥1 and real L,c. Its integer-parameter error is |c|/(n+d); convergence is not guessed from finitely many samples.

Representative formula (exact TeX)

u(n)

Assumptions

• N0 includes zero, and n+d>0 for every term. Exercises use safe-integer L,c,d, positive rational ε=p/q, and nonnegative safe-integer n and candidate N.
• All tail errors in this fixed model are nonincreasing, so a uniform cutoff N is valid exactly when |c|q<p(N+d). Equality fails the strict error condition.
• The least permitted cutoff is max(0,floor(|c|q/p)−d+1). At c=0, N=0 suffices; L does not affect error or cutoff.

Symbol scope

• u — specified real sequence
• n — natural index starting at zero

Intuition, definition, and misconceptions

The cutoff may depend on ε, but once chosen must control the entire tail. Checking one term or finitely many initial terms cannot prove convergence of an arbitrary sequence. This reciprocal model has a monotone error bound permitting the exact inequality.

Strict error changes the least cutoff: c=1, d=1, ε=1/2 makes the error at N=1 equal to ε, so the least valid cutoff is two. Positive and negative c have the same absolute error.

These checks concern the declared model and its exact parameter eligibility, not a new general convergence proof or formal bridge. Changing L does not give a new error calculation, and changing c,d does not give a new limit-value calculation.

Common errors include choosing one cutoff independent of every ε, checking only finitely many terms, or accepting error equal to ε.

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 — First fix the sequence and index convention.
• Tends-to relation — Tending to a value does not require attaining it.
• Quantifiers and explicit domains — Retain the order ∀ε, ∃N, ∀n≥N.
• Rational number — Represent positive tolerance and error exactly.

Verification boundary

The fixed-model parameters and tail bounds can be recomputed exactly; general real-sequence convergence has no newly attached formal Evidence.

not_formalized

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

Next obligation

Bind pinned Tendsto atTop, real neighborhoods, natural indexing, and the all-tail obligation for reciprocal cutoffs.

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.