Question: Let's suppose that $a\_n$ satisfy $|a\_{n+1}-a\_n|<2^{-n}$ for all $n$ . How can I prove that $a\_n$ is a cauchy sequence?

Answer: 
Indeed the graph is *bipartite* and 1 and 8 are on opposite *sides* (all edges go from one side to the other). Going from one to the other can only be done in an odd number of steps.
[![enter image description here](https://i.stack.imgur.com/GTbTQ.png)](https://i.stack.imgur.com/GTbTQ.png)

The graph can also be shown to be *layered*, so that the shortest path from 1 to 8 has length 3.

[![enter image description here](https://i.stack.imgur.com/XWIrN.png)](https://i.stack.imgur.com/XWIrN.png)
The cube graph is an example of a [bipartite graph](https://en.wikipedia.org/wiki/Bipartite_graph), where the vertices can be divided into two group such that the edges always connect a vertex from one group to a vertex in the other group, never to another vertex in the same group. 

The group can be seen in your diagram by moving vertex $1$ in amongst the other odd-numbered vertices and vertex $8$ likewise into the even-numbered group. Then you will always traverse an odd number of edges to reach a vertex in the other group from the starting vertex, plus all cycles are even length, etc.