I'm looking at some old question papers and here's a problem:
If $n$ and $m$ are integers and $d(n, m) = 2^{-r}$ if $|n - m| = 2^{r}t$ with $t$ odd if $n \neq m$ and $d_2(n, n) = 0$. Prove that the metric space $(\mathbb Z, d)$ is not complete. (Hint: Consider the sequence $a_n = \frac{4^{n} - 1}{3}$.)
The given hint doesn't really make sense to me, as every element of the form $a_n = \frac{4^n - 1}{3}$ can be shown to be odd. Then the distance between any two elements of $(a_n)$ would be $2^{-0} = 1$ according to the given metric $d(\cdot, \cdot)$. So this is not even a Cauchy sequence in $(\mathbb Z, d)$.
Edit: I realized that this is incorrect, after seeing the comments. $(a_n)$ is indeed a Cauchy sequence -- one should consider the difference between the elements of $(a_n)$.
Is there any other sensible Cauchy sequence that I can use to prove that this metric on $\mathbb Z$ is not complete?