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 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.
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.