Uniform Product of Continuous Functions is Continuous/Proof 1
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Theorem
Let $X$ be a metric space.
Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field.
Let $\sequence {f_n}$ be a sequence of bounded continuous mappings $f_n: X \to \mathbb K$.
Let the product $\ds \prod_{n \mathop = 1}^\infty f_n$ converge uniformly to $f$.
Then $f$ is continuous.
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 the Uniform Limit Theorem, $\ds \prod_{n \mathop = n_0}^\infty f_n$ is continuous.
Because $f_1, \ldots, f_{n_0 - 1}$ are continuous, so is $\ds \prod_{n \mathop = 1}^\infty f_n$.
$\blacksquare$