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: 
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.
As you can see you have bipartite graph here and one of the things of bipartite graph is that for any u,v vertices all the paths from u to v has to odd or even (can't be 1 path odd while there is other even) so if we have bipartite graph and we show 1 odd path then all the paths are odd.