LemmaStatement: AI-adaptedProof: AI-generatedprecheck 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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 $α, β \in R$ ,

$α f_{k} + β g_{k} ⟶ α f + β g$

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 $α, β$ .

[A1]

Uniform convergence gives, for every real $η > 0$ , one index after which $∣ f_{k} (x) - f (x) ∣ < η$ 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 $u, v$ , $∣ u + v ∣ \leq ∣ u ∣ + ∣ v ∣$ , while $∣ c u ∣ = ∣ c ∣ ∣ u ∣$ and absolute values are nonnegative (The triangle inequality, Basic properties of the absolute value).

Proof

technique · direct

1.1

Let $ε > 0$ be real and put $η : = ε / (2 (∣ α ∣ + ∣ β ∣ + 1)) > 0$ .

constructL1

1.2

By uniform convergence choose $N_{f}, N_{g}$ such that $∣ f_{k} (x) - f (x) ∣ < η$ for $k \geq N_{f}$ and all $x$ , and $∣ g_{k} (x) - g (x) ∣ < η$ for $k \geq N_{g}$ and all $x$ .

A1choose

2.1

Choose an index $N$ at least as large as $N_{f}$ and $N_{g}$ . For every $k \geq N$ and $x \in X$ , one has $∣ α (f_{k} (x) - f (x)) + β (g_{k} (x) - g (x)) ∣ \leq ∣ α ∣ η + ∣ β ∣ η < ε$ .

step 1.1step 1.2L1choosealgebra

3.1

The expression in step 2.1 is $∣ (α f_{k} + β g_{k}) (x) - (α f + β g) (x) ∣$ , and the index $N$ serves every $x$ , so the asserted convergence is uniform.

step 2.1A1∎

Depends on

Used by

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Sources

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