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.
The 8 vertices of the graph can be divided into 2 disjoint sets $V\_1=\{1,3,5,7\}$ and $V\_2=\{2,4,6,8\}$. Each edge of the graph connects a vertex of $V\_1$ to a vertex of $V\_2$, therefore, moving from a point of one of the sets to another point of the same set requires an even number of steps, and moving from one set to another requires an odd number of steps. $1\in V\_1$ and $8\in V\_2$, so the trip from one to another cannot be made in an even number of steps.