Given 2 sequences of functions $f_n$ and $g_n$ on an interval $[a, b]$.
$f_n$ is uniformly convergent to $f$.
$g_n$ is uniformly convergent to $g$.
And there exists 2 real numbers $M_1$ and $M_2$ such that:
$|f(x)| < M_1$
$|g(x)| < M_2$
for any $x \in [a,b]$.
How would I prove the the product $f_n \cdot g_n$ converges uniformly to $f\cdot g$?
I think the product does not converge uniformly, but having trouble proving it.