Prove convergence of recursive sequence

Question

I have tried to solve this question in a few ways but can't reach a final solution any way I try.
Question :
Let $x_{n+1}=10x_n^2$ be a sequence.
Prove the sequence converges $\forall x_0 \in [-0.1,0.1]$ and diverges $\forall x_0 \notin [-0.1,0.1]$
My try :
Assume $L\in \mathbb{R}$ be a limit of $x_n$.
Then : $L=10L^2 \longrightarrow L(10L-1)=0$
the limit can either be $L=0$ or $L=\frac{1}{10}$ From here on I don't know how to proceed