ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Translated tent functions on $R$ converge to zero in the compact-open topology

Example

For $k \in N$ , define

$f_{k} (x) : = \max {1 - ∣ x - k ∣, 0} (x \in R) .$

Then each $f_{k}$ is continuous and $1$ -Lipschitz, and $f_{k} \to 0$ in the compact-open topology on $C (R, R)$ , but the sequence does not converge uniformly on $R$ .

Facts & Assumptions

Given: The translated tent functions $(f_{k})$ on the real line.

[L1]

The general and published metric-domain compact-open topologies agree (The general compact-open topology agrees with the published metric-domain definition).

[L2]

For metric domain and target, compact-open convergence is compact convergence (For a metric domain and a metric target the compact-open topology on $C (X, Y)$ is the topology of compact convergence).

[L3]

Every compact subset of $R$ is bounded (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[L4]

The Archimedean property provides a natural number larger than any prescribed real (Every complete ordered field is Archimedean).

[L5]

Uniform convergence requires one tail index to work at every point of the domain (Uniform convergence, and the topology of uniform convergence: the metric topology of the uniform metric on $Y^{X}$ and on $C (X, Y)$ ).

[L6]

General compact-open subbasic conditions test images of compact sets (The compact-open topology on $C (X, Y)$ for arbitrary topological spaces).

Verification

technique · direct

1.1algebra

The map $x \mapsto 1 - ∣ x - k ∣$ is $1$ -Lipschitz, and taking the maximum with $0$ preserves that bound. Hence every $f_{k}$ is continuous and $1$ -Lipschitz.

1.2L3L4

Let $K \subseteq R$ be compact. If $K = \emptyset$ , convergence on $K$ is vacuous. Otherwise [L3] gives $R > 0$ with $∣ x ∣ \leq R$ for $x \in K$ , and [L4] gives $N \in N$ with $N > R + 1$ .

2.1step 1.2

If $k \geq N$ and $x \in K$ , then $∣ x - k ∣ \geq k - ∣ x ∣ > 1$ , so $f_{k} (x) = 0$ . Thus the sequence is eventually identically zero on every compact $K$ , and therefore converges to zero uniformly on each compact set.

3.1L1L2L6step 2.1

By [L1], [L2], and [L6], step 2.1 is convergence in the general compact-open topology.

4.1L5∎

Yet $f_{k} (k) = 1$ for every $k$ , so no tail has $∣ f_{k} (x) ∣ < 1 / 2$ for every $x \in R$ . By [L5], convergence is not uniform on the whole real line.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

56 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

The Ascoli–Arzelà Theorem, BBT (standard reference, not scraped)

Translated tent functions on RR converge to zero in the compact-open topology