Uniform Product of Continuous Functions is Continuous/Proof 1

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 the Uniform Limit Theorem, $\displaystyle \prod_{n \mathop = n_0}^\infty f_n$ is continuous.

Because $f_1,\ldots,f_{n_0-1}$ are continuous, so is $\displaystyle \prod_{n \mathop = 1}^\infty f_n$.