You are aware of how this behaves?

$u_0=-1 \Rightarrow \displaystyle{\lim_{n \to \infty}}=-1$

$u_0 \neq -1 \Rightarrow \displaystyle{\lim_{n \to \infty}}=3$

Proving the first statement by induction should be pretty trivial.

You seem to know enough of what's going on to take a stab at proving the second statement.