Let $F =\left\{ f\in C^1[a,b] : || f|| \leq c, ||f'||\leq k\right\}$ and
$G = \left\{f\in C[a,b] : || f|| \leq c, f \text{ is globally Lipschitz with Lipschitz constant }\leq k \right\}$
where $a,b,c,k$ are fixed real constants, and the norm is the "sup" norm.
Then, the claim is that the closure of $F$ in $C([a,b])$, $\overline F$, is $G$.
I was able to show that $\overline F \subset G$, but I have no idea how to show the converse. I think that the same question was already asked( Every $K$-Lipschitz function can be uniformly approximated by $C^1$ functions with derivative bounded by $K$), but is there a way to show what I want without using absolute continuity/mollifers?