The following is a step in a proof: $f$ is a Lipschitz function from $E$ to $F$ where $E$ is a finite-dimensional Banach space and $F$ an arbitrary Banach space. $\phi\geq 0$ is a $C^\infty$ function with compact support, $\int\phi=1$, $\phi(x)=\phi(-x)$. The function $g(z)=\int f(z+x)\phi(x)dx$ is defined. The claim is that $||g||_{\text{Lip}}\leq||f||_{\text{Lip}}$.
I see why this would be true with some additional condition on $\phi$ (say, $||\phi||_{L^2}\leq 1$), but it's not clear to me why the given conditions are sufficient to control the Lipschitz constant of $g$.