Uniformly Convergent Product Satisfies Uniform Cauchy Criterion

Theorem

Let $X$ be a compact topological space.

Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field.

Let $\sequence {f_n}$ be a sequence of continuous mappings $f_n: X \to \mathbb K$.

Let the infinite product $\ds \prod_{n \mathop = 1}^\infty f_n$ converge uniformly on $X$.

Then $\ds \prod_{n \mathop = 1}^\infty f_n$ satisfies the uniform Cauchy condition for products.

Proof

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Also see

• Tail of Uniformly Convergent Product Converges Uniformly to One
• Factors in Uniformly Convergent Product Converge Uniformly to One