= Solution
{c}
For real-valued <functions>, the <Uniform Cauchy criterion> applied to <partial sums> says that $\sum u_n$ has <uniform convergence> exactly when
$$
\forall\varepsilon>0\ \exists N\ \forall q>p\geq N\ \forall x\in E:\quad\left|\sum_{n=p+1}^q u_n(x)\right|<\varepsilon.
$$
Sufficiency uses <completeness of the real numbers>.
Let $M$ be a common bound for the <uniformly bounded family of functions> $(g_n)$; increase it if necessary so that $M>0$. The assumed <uniform absolute convergence> of $\sum f_n$ gives $N$ such that
$$
\sum_{n=p+1}^q|f_n(x)|<\frac{\varepsilon}{M+1}
\qquad(q>p\geq N,\ x\in E).
$$
The <triangle inequality> now yields
$$
\boxed{\left|\sum_{n=p+1}^q f_n(x)g_n(x)\right|\leq M\sum_{n=p+1}^q|f_n(x)|<\varepsilon.}
$$
The <Uniform Cauchy criterion> therefore proves \b[<uniform convergence> of $\sum f_ng_n$.] The same estimate with $|f_ng_n|$ proves <uniform absolute convergence> as well.
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