For integers $a$ and $b$, define $d(a, b)$ to be 0 if $a=b$, or $\frac{1}{2^{n}}$ if $a \neq b$ and $n$ is the largest non-negative integer such that $a-b$ is a multiple of $2^{n}$. Show that $d$ is a metric on the integers $\mathbb{Z}$.

Does the sequence $x_{n}=2^{n}-1$ converge in this metric?