Partial Products of Uniformly Convergent Product Converge Uniformly

Definition
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 product $\ds \prod_{n \mathop = 1}^\infty f_n$ converge uniformly.

Then the sequence of partial products converges uniformly.

Proof
Let $n_0 \in \N$ be such that the sequence of partial products of $\ds \prod_{n \mathop = n_0}^\infty f_n$ converges uniformly.

By Product of Bounded Functions is Bounded, $\ds \prod_{n \mathop = 1}^{n_0 - 1} f_n$ is bounded.

By Uniformly Convergent Sequence Multiplied with Function, the sequence of partial products converges uniformly.

Also see

 * Uniform Product of Continuous Functions is Continuous
 * Infinite Product of Analytic Functions is Analytic