Let $(f_n),(g_n)$ are sequences of functions on $\mathbb R$.
I was wondering when $ (f_n),(g_n) $ are uniformly convergent then what the conditions that can make the following statement is true?
$\lim\limits_n (f_n(x).g_n(x) )= \lim\limits_n f_n(x) .\lim\limits_n g_n(x)$ (Uniformly)
I know that this statement isn't always true. For example; we can take $f,g:[0,\infty) \to \mathbb R, f_n(x)= \frac{1}{n}, g_n(x)=x$ are uniformly convergent but $(f_n.g_n)=\frac{x}{n}$ isn't uniformly convergent.
Also, it is known that it is true when this two sequence are uniformly bounded but is it true under another conditions?
For example if we take $f_n,g_n: (0,1) \to \mathbb R$
$f_n(x)=\frac{1}{x} $ and $g_n(x)= \frac{1}{x^2}$
then
$$\lim\limits_n (f_n(x).g_n(x) )= \frac{1}{x^3}= \lim\limits_n f_n(x) .\lim\limits_n g_n(x)$$ (Uniformly)
But both of these sequences are not even bounded functions. So I feel like there must be a more strong condition that it will include the simple cases like this one. I can't find it so I am wondering if any one has read a condition like that.
Thank you