Let $N$ be a normed vector space, let $M$ be a metric space, and let $A$ be a subset of $M$. Let $V$ denote the space of all bounded functions from $A$ to $N$, with the sup norm. Let $C_b$ denote the set of all bounded continuous functions from $A$ to $N$. Show that $C_b$ is closed in $V$.
I think I'm suppose to use Arzela-Ascoli Theorem but do not know how to go about it