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.

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 $B,C\ge0$ such that

$|f(x)|\le B\quad\text{and}\quad |g_k(x)|\le C$

for every $x\in X$ and every $k\in\mathbb{N}$ . Then $f_kg_k\to fg$ 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 $|f(x)|\le B$ and $|g_k(x)|\le C$ 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 $\mathbb{R}$ 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 $u,v,c$ , $|u+v|\le|u|+|v|$ and $|cu|=|c||u|$ (The triangle inequality, Basic properties of the absolute value).

Proof

technique · direct

1.1

Let $\varepsilon>0$ be real and put $\eta:=\varepsilon/(B+C+1)>0$ .

construct

1.2

Choose $N$ such that, for every $k\ge N$ and every $x\in X$ , both $|f_k(x)-f(x)|<\eta$ and $|g_k(x)-g(x)|<\eta$ .

A1choose

2.1

For $k\ge N$ and $x\in X$ , add and subtract $f(x)g_k(x)$ to obtain $|f_k(x)g_k(x)-f(x)g(x)|\le |g_k(x)|\,|f_k(x)-f(x)|+|f(x)|\,|g_k(x)-g(x)|<(B+C)\eta<\varepsilon$ .

step 1.1step 1.2A2L1algebra

3.1

Since $N$ is independent of $x$ , step 2.1 proves $f_kg_k\to fg$ 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 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)