Absolute Value of Uniformly Convergent Product

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Theorem

Let $X$ be a compact topological space.

Let $\mathbb K$ be a field with absolute value $\left\vert{\, \cdot \, }\right\vert$.

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

Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge uniformly to $f$.


Then $\displaystyle \prod_{n \mathop = 1}^\infty \left\vert{f_n}\right\vert$ converges uniformly to $ \left\vert{f}\right\vert$.


Proof


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