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 $\left\langle{f_n}\right\rangle$ be a sequence of bounded mappings $f_n:X\to \mathbb K$.

Let the product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ 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 $\displaystyle \prod_{n \mathop = n_0}^\infty f_n$ converges uniformly.

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