If $f_n$ and $g_n$ are bounded sequences of functions that converge uniformly on a set $E$, show that $f_ng_n$ converges uniformly on $E$.
What I've tried
Call the functions that $f_n,g_n$ converge to $f,g$, respectively.
My intuition is that normalizing the functions here will help with the argument.
To simplify the notation, I will refer to $f(x)$ as $f$.
We need to show that:
$$\sup_{x\in E} \left| f_ng_n - fg \right| \rightarrow 0$$
We will deal with two cases, first assume that $f,g$ are bounded away from $0$ so that the above is equivalent to:
$$\sup_{x\in E} \left| \frac{f_n}{f}\frac{g_n}{g} - 1 \right| \rightarrow 0$$
factoring out $\frac{f_n}{f}$ we have that the term inside the $\sup$ is:
$$\left |\frac{f_n}{f} \left(\frac{g_n}{g} - 1\right) + \left(\frac{f_n}{f} - 1\right)\right |$$
By the triangle inequality it is sufficient to show that each of these converges to $0$ uniformly. The second term converges uniformly to $0$ from the hypothesis. The first term is:
$$O(1)\left(\frac{g_n}{g} - 1\right)$$
since $f_n$ is bounded and $f$ is bounded away from $0$. This converges uniformly to $0$ by the hypothesis on $g_n$.
Now I'm stuck figuring out how to handle the case where $f,g$ may be arbitrarily close to $0$ for some $x$. Is there some way to complete this argument, or does splitting it into these two cases not make sense here?
Hint: $f_ng_n -fg = (f_ng_n -fg_n) + (fg_n -fg).$