CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-08-10 (gpt-5.6-terra-codex-subscription)

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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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The uniform limit of uniformly continuous real-valued functions is uniformly continuous

Statement

Let $(X, d)$ be a metric space. If each $f_{k} : X \to R$ is uniformly continuous and $f_{k} \to f$ uniformly on $X$ , then $f$ is uniformly continuous.

Facts & Assumptions

Given: A metric space $(X, d)$ , uniformly continuous functions $f_{k} : X \to R$ , and uniform convergence $f_{k} \to f$ .

[A1]

Uniform convergence gives one index serving every point for any prescribed positive real error (Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions).

[A2]

Uniform continuity of $f_{N}$ means that for every real $η > 0$ there is $δ > 0$ such that $d (x, y) < δ$ implies $∣ f_{N} (x) - f_{N} (y) ∣ < η$ for all $x, y \in X$ (Uniform continuity of a map of metric spaces: one $δ$ serving every point, The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded).

[L1]

For reals $u, v$ , $∣ u + v ∣ \leq ∣ u ∣ + ∣ v ∣$ (The triangle inequality).

Proof

technique · direct

1.1

Let $ε > 0$ be real. Choose $N$ such that $∣ f_{N} (x) - f (x) ∣ < ε / 3$ for every $x \in X$ .

A1choose

1.2

By uniform continuity of $f_{N}$ , choose $δ > 0$ such that $d (x, y) < δ$ implies $∣ f_{N} (x) - f_{N} (y) ∣ < ε / 3$ for every $x, y \in X$ .

A2choose

2.1

If $d (x, y) < δ$ , then $∣ f (x) - f (y) ∣ \leq ∣ f (x) - f_{N} (x) ∣ + ∣ f_{N} (x) - f_{N} (y) ∣ + ∣ f_{N} (y) - f (y) ∣ < ε$ .

step 1.1step 1.2L1algebra

3.1

The same $δ$ serves every pair $x, y$ , so $f$ is uniformly continuous.

step 2.1A2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 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

J. Lebl, Basic Analysis I, §6.1 (standard reference, not scraped)
W. Trench, Introduction to Real Analysis (standard reference, not scraped)