Is a set consisting of the terms of a convergent sequence in $\mathscr C_b(A, \mathbb R^m)$ closed in $\mathscr C_b(A, \mathbb R^m)$?

Question

This is a follow-up to my previous question:

Let $f_k$ be a convergent sequence in $\mathscr C_b(A, \mathbb R^m)$.

Is $S=\{f_k \mid k = 1,2,\ldots\}$ closed in $\mathscr C_b(A, \mathbb R^m)$?

Obviously, $S$ is closed iff. the limit function (call it $f$) lies in S. But is the given information sufficient to answer the question conclusively?