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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.

The exponential series gives $x^x=e^{x\log x}={\sum }_{k=0}^{\infty }(x\log x{)}^k/k!$. Since $|x\log x|\le 1/e$ on $(0,1)$, the Weierstrass M-test proves uniform absolute convergence, using the summable bounds $e^{-k}/k!$. Therefore one can integrate the series term by term over this finite interval. Using part i,This is the alternating identity in sophomore's dream. The same estimate also proves absolute convergence of the integrated series; no cancellation is needed to justify the interchange.

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.