Let $\displaystyle (f_n)$ be a sequence of functions that converges uniformly to $\displaystyle f$ on A and that satisfies $\displaystyle | f_n (x) | \le M$ for all $\displaystyle n \in N$ and all $\displaystyle x \in A$.

If $\displaystyle g$ is continuous on the interval $\displaystyle [-M,M], $show that the sequence $\displaystyle (g \circ f_n )$ converges uniformly to $\displaystyle g \circ f$ on $\displaystyle A$.

Can anyone give me some hints to start the proof?