Density of Lipschitz functions with compact support

Question

Let $(X,d)$ be a locally compact metric space. Consider the space of complex continuous functions $C_0(X)$ which vanish at infinity.

I am interested in the following fact:

The space of compactly supported Lipschitz functions $Lip_{c,d}(X)$ is uniformly dense in $C_0(X)$.

I know that $Lip_d(X)$ is uniformly dense in $C_0(X)$, and that $C_c(X)$ is uniformly dense in $C_0(X)$.

1. But how can one prove that $Lip_{c,d}(X)$ is uniformly dense in $C_c(X)$?
2. Is there any reference for the above fact?

Many thanks!