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)
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.