CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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.
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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 $f_k:X\to\mathbb{R}$ 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 $f_k:X\to\mathbb{R}$ , 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 $\eta>0$ there is $\delta>0$ such that $d(x,y)<\delta$ implies $|f_N(x)-f_N(y)|<\eta$ for all $x,y\in X$ (Uniform continuity of a map of metric spaces: one $\delta$ serving every point, The absolute value makes $\mathbb{R}$ 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 $u,v$ , $|u+v|\le|u|+|v|$ (The triangle inequality).

Proof

technique · direct

1.1

Let $\varepsilon>0$ be real. Choose $N$ such that $|f_N(x)-f(x)|<\varepsilon/3$ for every $x\in X$ .

A1choose

1.2

By uniform continuity of $f_N$ , choose $\delta>0$ such that $d(x,y)<\delta$ implies $|f_N(x)-f_N(y)|<\varepsilon/3$ for every $x,y\in X$ .

A2choose

2.1

step 1.1step 1.2L1algebra

3.1

The same $\delta$ 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 40 results over 11 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

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