This is number 2 of the problem set for real analysis problems:
Prove that if
$\displaystyle
\sum_{n=1}^{\infty}{{g_n}}
$
converges uniformly, then (g_n) converges uniformly to zero.
If $\displaystyle \sum_{n=1}^{\infty} g_n$ converges uniformly then it is uniformly Cauchy.
Let $\displaystyle \epsilon > 0$ then there is $\displaystyle N$ so that if $\displaystyle n,m>N$
Then, $\displaystyle \left| \sum_{k=1}^n g_k(x) - \sum_{k=1}^m g_k(x) \right| < \epsilon$ for $\displaystyle x\in S$.
Thus, for example if $\displaystyle n=m+1$ then, $\displaystyle |g_n(x)| < \epsilon$.
Thus, we see that $\displaystyle g_n \to 0$ uniformly on $\displaystyle S$.