# Exact supremum error on a three-point domain [← Guide index](https://ocme.evemisslab.com/en/derivations/) · [Original MKO](https://ocme.evemisslab.com/en/mko/mko-uniform-convergence-of-functions/) This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented. derivation-mko-uniform-convergence-of-functions-01 · ocme-new142-reader-adoption-20261021 ## Goal and scope Prove g_n(a)=1/(n+2),g_n(b)=-4/(n+2),g_n(c)=2/(n+2) converges uniformly to zero. A complete finite domain and rational tails for all n≥0; N must not depend on x. ## Givens and symbols - algebra - epsilon - definition_here - G1 — D={a,b,c}, g=0; the three formulas define all of g_n:D to R for every n≥0. - E_n — whole-domain supremum error - N — common tail index ## Rules and sources Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged. - mko_definition · cited — 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. · mko-uniform-convergence-of-functions - algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities. - epsilon · prerequisite — Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals. - definition_here · prerequisite — A nonempty finite set of real numbers has its maximum as its supremum; uniform convergence requires, for each positive epsilon, one common N making the error smaller than epsilon for every n≥N and every x. ## Main derivation S01 · impliesBefore n\ge0 After n+2>0 Conditions: none Rule: algebra · Uses: G1 Leaf basis: algebra All points share a positive denominator, so absolute-error calculations avoid division by zero. S02 · equivalentBefore (|g_n(a)|,|g_n(b)|,|g_n(c)|) After (1,4,2)/(n+2) Conditions: none Rule: algebra · Uses: G1, S01 Leaf basis: algebra Take each absolute value, retaining magnitude 4 at b. S03 · definitionBefore E_n=\sup_{x\in D}|g_n(x)-g(x)| After E_n=\max\{1,4,2\}/(n+2)=4/(n+2) Conditions: none Rule: definition_here · Uses: S02 Leaf basis: definition_here The complete finite domain makes the supremum the maximum of its three errors, attained at b. S04 · impliesBefore \varepsilon>0,\quad N=\lfloor4/\varepsilon\rfloor+1 After N>4/\varepsilon Conditions: none Rule: epsilon · Uses: S03 Leaf basis: epsilon An Archimedean integer choice gives N depending only on epsilon. S05 · impliesBefore n\ge N After |g_n(x)|\le E_n=4/(n+2)<\varepsilon\quad\forall x\in D Conditions: x\in D,\ n\ge N Rule: algebra · Uses: S03, S04 Leaf basis: algebra Every later denominator exceeds 4/epsilon and simultaneously controls all three points. S06 · definitionBefore \forall\varepsilon>0\ \exists N\ \forall n\ge N\ \forall x\in D After g_n\to0\text{ uniformly on }D Conditions: none Rule: definition_here · Uses: S05 Leaf basis: definition_here The quantifier order is exactly uniform convergence with a common N. S07 · impliesBefore N_x=\lfloor |c_x|/\varepsilon\rfloor+1,\quad(c_a,c_b,c_c)=(1,-4,2) After \max_{x\in D}N_x=\lfloor4/\varepsilon\rfloor+1=N Conditions: none Rule: epsilon · Uses: G1 Leaf basis: epsilon The finite-maximum proof agrees with the exact-supremum certificate. S08 · impliesBefore \varepsilon=1/10,\quad N=41 After E_n\le4/43<1/10\quad(n\ge41) Conditions: none Rule: epsilon · Uses: S04, S05 Leaf basis: epsilon Substitute one rational epsilon as an arithmetic audit; the symbolic tail already proves the general conclusion. ## Substeps and lemmas No substeps in this guide. ## Reverse audits - S08 · premise_audit · S04, S05 — This numerical audit does not replace the proof for every epsilon. - S07 · premise_audit · G1 — The maximum exists because the entire domain is finite. - S06 · premise_audit · S05 — Choosing N only after x would not prove general uniform convergence. - S05 · premise_audit · S03, S04 — The statement is for all n≥N, not only n=N. - S04 · premise_audit · S03 — The formula for N contains no x. - S03 · premise_audit · S02 — No sampled maximum is used to estimate an unknown infinite-domain supremum. - S02 · inverse_equivalence · G1, S01 — The maximum signed value is not the maximum absolute error. - S01 · premise_audit · G1 — The common index and domain remain fixed. ## Error controls and negative controls Absolute values firstList absolute errors before taking their maximum. Common indexN depends only on epsilon and controls every later n. - The supremum error is 2/(n+2). → reject — It misses b's absolute error. - N may depend on x with no common bound. → reject — That is only the pointwise quantifier order. - A finite-grid supremum must equal the supremum over the whole continuous interval. → reject — For h(x)=x*(1-x), grid {0,1} has maximum zero, while x=1/2 in [0,1] has value 1/4. ## Result and limitations The supremum error is exactly 4/(n+2); common N=floor(4/epsilon)+1 proves uniform convergence. ## Verification and source binding 1 declared checks for this guide; only the listed expressions are covered. - source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f - adopted source SHA-256: 1a3da85dbb40310b278c7800673a12807c5d786870d07ca8dc757999c5f695d4 - original source SHA-256: 729e76f868ad91fc9a80212ab0f92d3ee8fb3af4d052e2d03b2201388c4fd253 - Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW [Same-source full JSON (formulas, ASTs, controls and revision log)](https://ocme.evemisslab.com/data/derivations/coverage-20261021/guides/derivation-mko-uniform-convergence-of-functions-01.json)