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.

Products converge uniformly when both factors converge uniformly and one limiting factor and one approximating family are uniformly bounded

Statement

Let $X$ be a set, and suppose $f_{k} \to f$ and $g_{k} \to g$ uniformly on $X$ . Assume there are reals $B, C \geq 0$ such that

$∣ f (x) ∣ \leq B and ∣ g_{k} (x) ∣ \leq C$

for every $x \in X$ and every $k \in N$ . Then $f_{k} g_{k} \to f g$ uniformly on $X$ .

The same conclusion holds after interchanging the two factors: it is enough that one limit function and the approximating sequence of the other factor have uniform bounds.

Facts & Assumptions

Given: Uniform convergence $f_{k} \to f$ and $g_{k} \to g$ on $X$ , with bounds $∣ f (x) ∣ \leq B$ and $∣ g_{k} (x) ∣ \leq C$ for all $x, k$ .

[A1]

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

[A2]

A subset of $R$ is bounded when it has real lower and upper bounds; the displayed absolute-value inequalities are the corresponding uniform bounds on the ranges (Lower bound, bounded below, bounded set).

[L1]

For reals $u, v, c$ , $∣ u + v ∣ \leq ∣ u ∣ + ∣ v ∣$ and $∣ c u ∣ = ∣ c ∣ ∣ u ∣$ (The triangle inequality, Basic properties of the absolute value).

Proof

technique · direct

1.1

Let $ε > 0$ be real and put $η : = ε / (B + C + 1) > 0$ .

construct

1.2

Choose $N$ such that, for every $k \geq N$ and every $x \in X$ , both $∣ f_{k} (x) - f (x) ∣ < η$ and $∣ g_{k} (x) - g (x) ∣ < η$ .

A1choose

2.1

For $k \geq N$ and $x \in X$ , add and subtract $f (x) g_{k} (x)$ to obtain $∣ f_{k} (x) g_{k} (x) - f (x) g (x) ∣ \leq ∣ g_{k} (x) ∣ ∣ f_{k} (x) - f (x) ∣ + ∣ f (x) ∣ ∣ g_{k} (x) - g (x) ∣ < (B + C) η < ε$ .

step 1.1step 1.2A2L1algebra

3.1

Since $N$ is independent of $x$ , step 2.1 proves $f_{k} g_{k} \to f g$ uniformly. Interchanging the names of the factors gives the symmetric clause.

step 2.1A1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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)