If $f \in\operatorname{Lip}_K[a, b]$, show that $f$ can be uniformly approximated by polynomials in $\operatorname{Lip}_K[ a, b]$.
Context: $f \in \operatorname{Lip}_K[a,b]$ then it is Lipschitz with constant $K$. The text I am currently using is Real Analysis by Carothers. We have developed Stone-Weierstrass and have seen that $\operatorname{Lip}[a,b]=\cup\operatorname{Lip}_K[a,b]$ for $K\in\mathbb{N}$ is a subalgebra $C[a,b]$.
I was wondering if we could somehow adapt the proof of Weierstrass theorem or maybe I am missing some fact about polynomials on a interval that relates to Lipschitz.