TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passverified 2026-08-10 (gpt-5.6-terra-codex-subscription)

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Peano's form: the normalized Taylor remainder tends to zero

Statement

Let $n\ge1$ . If there is a real $\delta>0$ such that $f$ is $n$ -times differentiable on the open interval $N_\delta(a)=(a-\delta,a+\delta)$ , then $\frac{R_{n,a}f(x)}{(x-a)^n}\longrightarrow0\qquad(x\to a).$ Equivalently, in the usual little- $o$ shorthand, $f(x)=T_{n,a}f(x)+o((x-a)^n)$ . For $n=0$ , the analogous assertion is the separate continuity condition at $a$ : for every $\varepsilon>0$ , all domain points $x$ sufficiently near $a$ satisfy $|f(x)-f(a)|<\varepsilon$ .

Facts & Assumptions

Given: The stated differentiability on the open neighbourhood $N_\delta(a)$ of The $\varepsilon$ -neighbourhood and the punctured $\varepsilon$ -neighbourhood of a point of $\mathbb{R}$ , or the separate continuity hypothesis when $n=0$ .

[L1]

Taylor polynomials and their matching derivatives are Taylor polynomials and their remainders and Taylor polynomials match the prescribed derivatives at the centre.

[L2]

The derivative quotient is The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, differentiability implies continuity (A function differentiable at $c$ is continuous at $c$ ), and continuity at $a$ has the stated quantified condition (Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$ : the $\varepsilon$ - $\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point). The Cauchy quotient lemma is Cauchy's mean-value theorem in quotient form when the denominator derivative is nonzero, the shifted-power derivative follows from For a natural $n \ge 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $\iota(n)\,x^{\,n-1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \ge 1$ the function $x \mapsto x^{-n}$ is differentiable at every $x \ne 0$ with derivative $-\iota(n)\,x^{-n-1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term and The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)'(c) = f'(g(c))\,g'(c)$ , and finite limits are The $\varepsilon$ - $\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$ and obey Sums, scalar multiples, products and quotients of function limits, the quotient under the hypothesis that the denominator limit is nonzero.

[L3]

For $n\ge1$ , the canonical real $\iota(n)$ is positive and hence nonzero (The canonical natural $\iota(n) = n \cdot 1_F$ of a field, Canonical naturals are positive and strictly increasing).

Proof

technique · induction

1.1

Under the separate $n=0$ hypothesis, the quantified assertion is exactly the definition of continuity at $a$ .

givenL2

1.2

For $n=1$ , the derivative definition gives $\frac{f(x)-f(a)-f'(a)(x-a)}{x-a} =\frac{f(x)-f(a)}{x-a}-f'(a)\longrightarrow0.$ This is the required base case.

basegivenL2

1.3

Assume $n\ge2$ and the assertion through order $n-1$ . Put $R=R_{n,a}f$ . Then $R(a)=R'(a)=0$ , and $R'(x)=R_{n-1,a}(f')(x)$ .

L1algebra

2.1

By the induction hypothesis applied to $f'$ , $R'(x)/(x-a)^{n-1}\to0$ . Applying the Cauchy quotient lemma to $R(x)-R(a)$ and $(x-a)^n$ gives $R(x)/(x-a)^n=R'(\xi)/(\iota(n)(\xi-a)^{n-1})$ for a point $\xi$ between $a$ and $x$ .

step 1.3ihL2L3

3.1

Since $\xi\to a$ , the right side tends to $0$ . This proves the Peano estimate without assuming continuity of $f^{(n)}$ .

step 2.1L2discharge-induction∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 99 results over 25 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)
Taylor's theorem (Wikipedia): statement and Peano remainder (standard reference, not scraped)