= Solution
Write $v_2(m)$ for the <2-adic valuation> of a nonzero integer $m$. Then $d(a,b)=2^{-v_2(a-b)}$ for unequal arguments. It is positive for $a\ne b$, vanishes exactly when $a=b$, and is symmetric because $v_2(-m)=v_2(m)$.
For the <triangle inequality>, if $2^r$ divides $a-b$ and $2^s$ divides $b-c$, then $2^{\min(r,s)}$ divides their sum $a-c$. Thus
$$
d(a,c)\le\max\{d(a,b),d(b,c)\}\le d(a,b)+d(b,c).
$$
Coincident arguments also satisfy these inequalities immediately. Consequently \b[$d$ is a <metric>], and indeed an <ultrametric> induced by the <2-adic absolute value>.
For the sequence, $x_n-(-1)=2^n$, so
$$
\boxed{d(x_n,-1)=2^{-n}\longrightarrow0,\qquad x_n\longrightarrow-1.}
$$
The same sequence diverges in the usual Euclidean <metric>; the convergence here depends on the <2-adic absolute value>.
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