LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passverified 2026-08-10 (gpt-5.6-terra-codex-subscription)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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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Uniform limits respect sums and scalar multiples

Statement

Let $X$ be a set. Suppose $f_k\to f$ and $g_k\to g$ uniformly on $X$ , where all functions are real valued. Then, for all $\alpha,\beta\in\mathbb{R}$ ,

$\alpha f_k+\beta g_k\longrightarrow \alpha f+\beta g$

uniformly on $X$ . In particular, uniform convergence is preserved by sums, differences, and scalar multiples.

Facts & Assumptions

Given: A set $X$ , uniformly convergent sequences $f_k\to f$ and $g_k\to g$ , and reals $\alpha,\beta$ .

[A1]

Uniform convergence gives, for every real $\eta>0$ , one index after which $|f_k(x)-f(x)|<\eta$ at every $x$ , and likewise for $g_k\to g$ (Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions).

[L1]

For reals $u,v$ , $|u+v|\le |u|+|v|$ , while $|cu|=|c||u|$ and absolute values are nonnegative (The triangle inequality, Basic properties of the absolute value).

Proof

technique · direct

1.1

Let $\varepsilon>0$ be real and put $\eta:=\varepsilon/\bigl(2(|\alpha|+|\beta|+1)\bigr)>0$ .

constructL1

1.2

By uniform convergence choose $N_f,N_g$ such that $|f_k(x)-f(x)|<\eta$ for $k\ge N_f$ and all $x$ , and $|g_k(x)-g(x)|<\eta$ for $k\ge N_g$ and all $x$ .

A1choose

2.1

step 1.1step 1.2L1choosealgebra

3.1

The expression in step 2.1 is $|(\alpha f_k+\beta g_k)(x)-(\alpha f+\beta g)(x)|$ , and the index $N$ serves every $x$ , so the asserted convergence is uniform.

step 2.1A1∎

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: 22 results over 7 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)