Metric on Shift of Finite Type is Metric

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

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

Then $d_\theta$ is a metric on $X _\mathbf A$.

Well-defined
Since $x,y\in X_\mathbf A$ are sequences, we have:
 * $x\ne y$ $\exists i\in\Z : x_i\ne y_i$

Thus, for $x\neq y$:
 * $\max \set {n \in \N : x_i = y_i \text { for all } i \in \openint {-n} n}$

exists.

Therefore the mapping
 * $d _\theta : X_\mathbf A \times X_\mathbf A \to \R$

is well-defined.

M1, M3, M4
These follow directly from the definition.