Show that a sequence of functions has a uniformly convergent subsequence on (-1,1)

Question

I'm studying for my Real Analysis qualifying exam and I'm a little unsure of one question. I'm given that $f_k$ is a sequence of continuous functions $(-1,1) \to \Bbb R$ such that $|f_k| \leq M$ for all $x$ and $k$. Let

$$g_k(x) = \int_a^x f_k(t)dt.$$

I have to show that this sequence, $g_k$, has a uniformly convergent subsequence.

Now my first instinct is to try using Arzela-Ascoli, but I'm a little iffy on whether it's applicable since the domain is an open interval. So I was thinking maybe I could extend $g_k$ to $[-1,1]$, but I'm not sure if I can do that or how I would go about showing I can.

Is this the right direction? Any help is greatly appreciated.

Answer 1

Hint 1: Use the boundedness of $f_k$ to show that $g_k$ is uniformly continuous on $(-1,1)$.

Hint 2: Show that an uniformly continuous function on $(-1,1)$ can be extended to a continuous function on $[-1,1]$. (this is a more general phenomena: any uniformly continuous function can be extended to the completion of your space).