= Solution
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 $\min(r,s)-1$. Hence
$$
n_{\min}(x,z)\geq\min\{n_{\min}(x,y),n_{\min}(y,z)\},
$$
and therefore
$$
d(x,z)\leq\max\{d(x,y),d(y,z)\}\leq d(x,y)+d(y,z).
$$
\b[Thus $d$ is a <metric>; in fact, the stronger first inequality makes it an <ultrametric>.]
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