= Bounded multipliers preserve uniform absolute convergence
{title2=$\sum_n|f_n|\text{ uniformly convergent},\quad\sup_{n,x}|g_n(x)|<\infty$}
Suppose a <series> of <functions> $\sum f_n$ has <uniform absolute convergence> on $E$ and $(g_n)$ is a <uniformly bounded family of functions> there, with common bound $M$. The <triangle inequality> gives
$$
\left|\sum_{n=p+1}^qf_n(x)g_n(x)\right|\leq M\sum_{n=p+1}^q|f_n(x)|.
$$
The right-hand side tends uniformly to zero as $q>p\to\infty$. The <Uniform Cauchy criterion> proves <uniform convergence> of $\sum f_ng_n$. Applying the same estimate to $\sum|f_ng_n|$ proves <uniform absolute convergence> too.
The shared bound over both $n$ and $x$ matters: separate bounds for each $g_n$ can grow too fast. For example, the constant <functions> $f_n(x)=2^{-n}$ and $g_n(x)=2^n$ have a summable original <series> but products identically one.
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