I know that $\mathcal{C}^k$ for $k\geq 1$ is dense in the space of Lipschitz funcions. My question in fact is: If $\{f_n\}_{n\geq 0}\subset \mathcal{C}^k$ such that $f_n \to f$ where $f$ is only Lipschitz. If such sequence $\{f_n\}_{n\geq 0}$ can be found in such a way that all derivatives are bounded and $\sup_n \|f_n^{(k)}\|_{\infty}<\infty$. So that I can upperbound "things" by the derivatives, and still get convergence when I let $n$ go to infinity.
Thanks a lot for your help! :)