Factors in Uniformly Convergent Product Converge Uniformly to One

Theorem

Let $X$ be a set.

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

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

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

Then $f_n$ converges uniformly to $1$.

Proof

Follows directly from Uniformly Convergent Product Satisfies Uniform Cauchy Criterion.

$\blacksquare$

Also see

• Tail of Uniformly Convergent Product Converges Uniformly to One