Given $t>0,r>0$. Let $\{x_k\}$ be given by $x_0=1$
$$ x_{k+1}=x_k-2t(x_k+r\cdot{\rm sign}(x_k)).$$
I would like to prove that the sequence does not converge and does not admit $0$ as a cluster point. I tried to see the behavior of the sequence by coding it on Matlab and it was right (that the sequence does not converge, and does not admit $0$ as a cluster point). However, I could not process the proof as dealing with the sign function is not easy to me. Thanks a lot for your help.