(I write $\omega$ for the set $\{0,1,2,\ldots\}$.)
Let $X$ denote a metric space with metric $d$. Define a function $d^{\star} : X^\omega \times X^\omega \rightarrow [0,\infty]^\omega$ by writing $$d^\star_i(x,y) = \sup_{n,m \geq i} d(x_n,y_m)$$
Please comment if this definition is unclear.
Question. Does $d^\star$ have an accepted name, or notation? Do any authors use it?
Some Observations.
A sequence $x \in X^\omega$ is constant iff the sequence $d^\star(x,x)$ equals the sequence $0 \in [0,\infty]^\omega.$ It is Cauchy iff the sequence $d^\star(x,x)$ converges to $0 \in [0,\infty]$.
$d^\star$ inherits lots of nice properties from $d$. For example, $d^\star$ is symmetric, and it satisfies the "triangle inequality" $$(\forall x,y,k \in X^\omega)\quad d^\star(x,y) \leq d^\star(x,k) + d^\star(k,x),$$ where $\leq$ is to be interpreted in the sense of the product order on $[0,\infty]^\omega$.