The problem says:
Let $(f_n)$ be a uniformly convergent sequence of $E-$valued functions on X, and $F$ a Banach space. Suppose that $f_n(X)\subseteq D$ for all $n\in\mathbb{N}$ and $g: D\to F$ is uniformly continuous. Show that $(g\,\circ\,f_n)$ is uniformly convergent.
I think the problem is that I can roughly understand the problem and write some unstrict proof, but I'm confused with how to write it out strictly with the $\varepsilon-N$ language. And I also fail to see the purpose of the condition Banach space.
My current solution:
Simply used the definition: for any positive $\varepsilon$ there exists a positive $\delta$ in that forall $d(x,x')<\delta$, $\Vert g(x)-g(x')\Vert<\varepsilon$. Then use again the definition and we find $N$ such that forall $n\geq N$ and $x\in D$, $|f_n(x)-f(x)|<\delta$. Finally we choose this $N(\varepsilon)$ and can get the desired uniform continuity.