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.
Notice that every dot is connected to 3 other dots and these three dots are of different parity ( if first dot is even then other three will be odd and if first dot is odd then the other three will be even). So if you start with even you will go to odd in next step and similarly when you start from odd you will go to even in next step. 

Now just start from 1 and notice that:

step 1 -reached even, 

step 2-reached odd, 

step 3-reached even
.
.
.
.
Step odd-reached even.

Since you want to reach 8 which is even so you will have to take odd steps.