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: 
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.
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.