The exercise states that (Vn) is a decreasing sequence converging at $0$. And let (un) be as shown below
and then asks to prove that (un) is a Cauchy sequence and thus convergent.
The definition of a cauchy sequence is given as:
The proof is shown below: 
The areas i'm a bit confused on are underlined in pink
Firstly how would we write the sequence for u$n+m$?
In step 2: How did we get $n+m$ on top of sigma and how did we get $k=n+1$?
and lastly in step 3 why is the $(-1)$ to the power of $m-1$, shouldn't it be to the power of $n+1$ since $k=n+1$?