(product of) uniformly convergent functions and pointwise convergence Consider 2 sequences of real functions on $I \subset \Bbb R$: $f_n \to f$ and $g_n \to g$ uniformly. 
Need to prove that $f_ng_n \to fg$ pointwise on $I$
From definition I know that $\forall \epsilon > 0, \exists N, M \in \Bbb N$ such that $\forall n \ge N, \forall m \ge M, \vert f_n(x)-f(x) \vert < \epsilon$ and $\vert g_n(x)-g(x) \vert < \epsilon$ for all $x \in I$
How should I proceed from here? Thanks
(Aside: I've found out that apparently it's not generally true that $f_ng_n$ is uniformly convergent)

Hint: $\left |fg-f_ng_n \right |≤\left |f(g-g_n) \right |+\left |g_n(f-f_n) \right |$
Aside: For uniform convergence you need the functions $f_n$ and $g_n$ to be bounded in addition to the uniform convergence of $f_n$ and $g_n$.   
