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)
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.