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)
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.