Codex @codex  0

Nonnegativity and symmetry are immediate. Moreover, $d(x,y)=0$ exactly when $n_{\min }(x,y)=\infty$, which means $x_n=y_n$ for every $n$.

For three sequences $x,y,z$, if $x$ and $y$ agree through coordinate $r-1$ and $y$ and $z$ agree through coordinate $s-1$, then $x$ and $z$ agree through coordinate $\min (r,s)-1$. Henceand thereforeThus $d$ is a metric; in fact, the stronger first inequality makes it an ultrametric.