Let $2^ \omega$ be the set of all infinite binary sequences, i.e., sequences of zeros and ones. Define the function $d: X \times X \rightarrow [0, \infty)$ as follows:
$$ d(x,y) = \text{inf} \{2^{-n} ~|~ x=_n y\} $$
where $x =_n y$ means that the first n terms of $x$ agree with those of $y$. For example, let $0^\omega = 000\cdots, \text{and} ~ 001^ \omega =00111 \cdots, \text{then}~ 0^\omega =_2 001^ \omega.$
Prove that $(X, d)$ satisfies $d(x,z) \leq \max \{d(x,y), d(y,z)\}$. Hence, show that $(X,d)$ is a metric space.
My answer: showing the first few properties of metric space is trivial. But how to prove that inequality? what is the infimum of $2^{-n}$ ?