I am trying to prove that every sequence for which

$|a\_{n+1}-a\_{n}| \leq \frac{1}{2}|a\_{n}-a\_{n-1}|$

is true, is bounded and convergent.
I tried defining $b\_n = a\_n$ and $c\_n = a\_{n-1}$ and then use Stolz, but it didn't work. 

How could I prove this?