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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 givesThe right-hand side tends uniformly to zero as $q>p\to \infty$. The Uniform Cauchy criterion proves uniform convergence of $\sum f_n{g}_n$. Applying the same estimate to $\sum |f_n{g}_n|$ 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.

For real-valued functions, the Uniform Cauchy criterion applied to partial sums says that $\sum u_n$ has uniform convergence exactly whenSufficiency 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 thatThe triangle inequality now yieldsThe Uniform Cauchy criterion therefore proves uniform convergence of $\sum f_n{g}_n$. The same estimate with $|f_n{g}_n|$ proves uniform absolute convergence as well.

Use uniform convergence to choose $N$ with $|f_n(x)-f_N(x)|\le 1$ for all $n\ge N$ and $x\in E$; this follows, for example, from the Uniform Cauchy criterion. Since each $f_j$ is a bounded function, put $M_j={\sup }_E|f_j|<\infty$ for $1\le j\le N$. Thenfor every $n$ and $x$. Thus $(f_n)$ is a uniformly bounded family of functions. The finite initial segment is essential: uniform convergence directly controls only a tail of a sequence.