A question related to convergence property of proximal point algorithm
Solution 1:
This inequality alone does not show convergence. It only implies boundedness of $(x_k)$ and $\|x_{k+1}-x_k\|\to0$. Note that a constant sequence $x_k=x^0\ne x^*$ also satisfies the inequality.
If the inequality is fulfilled only for some $\alpha\in (0,1)$ then not much can be said: $x^*=0$ and $x_k = \left(\frac{2}{1+\alpha}\right)^k$ satisfy the inequality but $(x_k)$ is unbounded.