Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 2 / 3H / 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 $a-c$. ThusCoincident arguments also satisfy these inequalities immediately. Consequently $d$ is a metric, and indeed an ultrametric induced by the 2-adic absolute value.

For the sequence, $x_n-(-1)=2^n$, soThe same sequence diverges in the usual Euclidean metric; the convergence here depends on the 2-adic absolute value.