Space of Lipschitz Functions is Banach Space/Shift of Finite Type

Theorem
Let $\struct {X _\mathbf A, \sigma_\mathbf A}$ be a shift of finite type.

Let $\theta \in \openint 0 1$.

Let $F_\theta$ be the space of Lipschitz mappings on $X _\mathbf A$.

Let $\norm \cdot_\theta$ be the Lipschitz norm on $F_\theta$.

Then $\struct {F_\theta, \norm \cdot_\theta}$ is a Banach space.