Suppose that $f_n → f$uniformly on some set $E$ and that for each $n$, there exists $M_n$ such that
$$|f_n(x)| ≤ M\quad\text{for all }n=1,2,3...\text{ and all }x ∈ E.$$
Suppose $g$ is a continuous function on $[-M,M]$.
How to show that $g(f_n(x))$ converges uniformly to $g(f(x))$ on $E$?