CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Every endpoint chord is no longer than the arc: $∥ γ (b) - γ (a) ∥_{2} \leq L (γ)$

Statement

Let $γ : [a, b] \to R^{n}$ be a path, with $n \geq 1$ . For every $a \leq u \leq v \leq b$ ,

$∥ γ (v) - γ (u) ∥_{2} \leq L_{[u, v]} (γ ∣_{[u, v]}) .$

This includes $u = v$ , when both sides are zero. In particular a path of length zero is constant.

Facts & Assumptions

Given: The path $γ$ and $u \leq v$ .

[L1]

For $u < v$ , the partition with point set ${u, v}$ has polygonal length $∥ γ (v) - γ (u) ∥_{2}$ , and arc length is the supremum of all polygonal lengths (Paths in $R^{n}$ , inscribed polygonal sums, arc length as their supremum, and rectifiability).

[L2]

On a singleton parameter interval, arc length is defined to be $0$ (Paths in $R^{n}$ , inscribed polygonal sums, arc length as their supremum, and rectifiability).

Proof

technique · direct

1.1

If $u < v$ , [L1] says the chord length is one member of the set whose supremum is the arc length, so it is at most that supremum.

givenL1

1.2

If $u = v$ , the chord is the zero vector and [L2] makes both sides zero.

givenL2

2.1

If the whole path has length zero, applying steps 1.1--1.2 to every $u \leq v$ makes every chord zero; separation for the Euclidean norm gives $γ (u) = γ (v)$ , so $γ$ is constant.

step 1.1step 1.2L1∎

Depends on

Paths in $\mathbb{R}^n$, inscribed polygonal sums, arc length as their supremum, and rectifiability

Used by

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

T. M. Apostol, Mathematical Analysis, Section 6.10 (standard reference, not scraped)

Every endpoint chord is no longer than the arc: ∥γ(b)−γ(a)∥2≤L(γ)∥γ(b)−γ(a)∥2​≤L(γ)