Convergence of sum of alternating functions Let $f_n:[-1,1]\to[0,1]$ be continuous functions. Assume that for every $x\in [-1,1]$, $\{f_n(x)\}$ is decreasing and $\lim_{n\to\infty}f_n(x)=0$. Let $g_n(x)=\sum_{m=1}^{n}(-1)^mf_m(x)$. Show that $\{g_n(x)\}$ converges to some $g(x)\in \mathbb{R}$ for all $x\in[-1,1]$. In particular, show that $g$ is continuous on $[-1,1]$.
If we can show $\{g_n\}$ converges uniformly to $g$, then we are done, right? However due to the way this problem is phrased, I think it's hard to directly show uniform convergence. But even if we can show point-wise convergence first, since $\{g_n\}$ aren't monotonic, we can't use Dini's theorem to claim $g$ is continuous.
Please help!

I have deleted my first proof because it required some extra work.
Simply follow the proof of the alternating series theorem in baby Rudin. The fact that $f_n$ decreases to $0$ uniformly will tell you that the series is uniformly convergent (by the same proof). [$f_n$ decreases to $0$ uniformly by Dini's Theorem]. 
