LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-21

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Continuous $R^{n}$ -valued curves on a nonempty compact interval form a complete supremum-metric space

Statement

Let $J$ be a nonempty compact interval and $n \geq 1$ . Continuous $R^{n}$ -valued curves on $J$ are complete in the supremum metric

$d_{\infty} (x, y) : = \sup_{t \in J} ∥ x (t) - y (t) ∥_{2} .$

Facts & Assumptions

Given: A $d_{\infty}$ -Cauchy sequence $(x_{m})$ in $C (J, R^{n})$ .

[L1]

For a nonempty compact metric space $K$ , $C (K, R)$ is complete in the supremum metric ( $C (K, R)$ is complete in the supremum metric for every nonempty compact metric space $K$ ).

[L2]

Any two norms on $R^{n}$ , $n \geq 1$ , are equivalent (For $n \geq 1$ all norms on $R^{n}$ are equivalent).

Proof

technique · direct

1.1givenL1

Each coordinate sequence is supremum-Cauchy, so [L1] gives a continuous coordinate limit; assembling the finitely many coordinate limits defines a continuous curve $x : J \to R^{n}$ .

2.1step 1.1L2algebra∎

The maximum-coordinate errors tend uniformly to zero, and [L2] bounds the Euclidean norm by a constant multiple of the maximum norm; hence $d_{\infty} (x_{m}, x) \to 0$ , including when $J$ is a one-point interval.

Continuous $R^{n}$ -valued curves on a nonempty compact interval form a complete supremum-metric space

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Continuous RnRn-valued curves on a nonempty compact interval form a complete supremum-metric space

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Continuous $R^{n}$ -valued curves on a nonempty compact interval form a complete supremum-metric space