show that $\mathbb{Z}$ is totally disconnected Show that $(\mathbb{Z},d)$ is totally disconnected (where $d$ is the metric induced by the Euclidean metric on $\mathbb{R}$).

I think that to prove this I should use contradiction but not really sure where to start
Hint: can you find neighbourhoods around each $x \in \mathbb Z$ such that the neighbourhoods are completely disjoint?
Note that a space is totally disconnected if its points are open, and in this case $\mathbb Z$'s points are open.  You can prove that by noting that $(n-1/2,n+1/2)\cap\mathbb Z=\{n\}$ for each $n\in\mathbb Z$.