= Solution
{c}
Let $S_N=\sum_{n=1}^Nf_n$ and let $s$ be its <uniform limit>. Choose a finite $B\geq0$ with $|g(x)|\leq B$ for all $x\in E$. The <partial sums> of the multiplied <series> are $gS_N$, and
$$
|g(x)S_N(x)-g(x)s(x)|\leq B|S_N(x)-s(x)|.
$$
For any $\varepsilon>0$, <uniform convergence> supplies $N_0$ making $|S_N(x)-s(x)|<\varepsilon/(B+1)$ for all $N\geq N_0$ and $x\in E$. The displayed estimate then gives an error less than $\varepsilon$. Thus
$$
\boxed{\sum_{n=1}^{\infty}f_n(x)g(x)=g(x)s(x)\quad\text{uniformly on }E.}
$$
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