I learned in my real analysis class that if $f_n:[a,b] \to \mathbb{R}$ is a sequence of differentiable functions such that $f_n \to f$ uniformly and $f_{n}^\prime \to g$ uniformly then $f$ is differentiable and f' = g.
Can the assumptions be weakened? In particular I'd like to dispense of the condition that $f_{n}^\prime$ converges uniformly to some function $g$.
This question is motivated by the following: Let K = \{f:[a,b] \to \mathbb{R}: f \text{ is differentiable and } \|f\|_{\infty} + \|f'\|_{\infty} \le 1\}. By the Arzelà-Ascoli theorem, the closure of this set is compact in $C([a,b])$. But I want it to actually be compact possibly after imposing some additional conditions on the possible functions $f$. This happens when $K$ is closed, hence the question.
Thanks in advance.