Let $(f_n)$ be a sequence of functions that converges uniformly to $f$ on A and that satisfies $| f_n (x) | \le M$ for all $n \in N$ and all $x \in A$.
If $g$ is continuous on the interval $[-M,M],$show that the sequence $(g \circ f_n )$ converges uniformly to $g \circ f$ on $A$.
Notice that g is uniformly continuous (as a continuous function on a bounded interval is uniformly continuous). Pick an epsilon, then $|g(f_n(x))-g(f(x))|<\epsilon$ for some $\delta$ such that $|f_n(x)-f(x)|<\delta$. Now what does uniform convergence of the f_n tell you about large enough n?