Suppose (f) satisfies a sharp growth condition around the solution set.
Then one can lower bound the step size of the proximal point iterates before termination.
Let [ x_{k+1} \in \arg\min_x \left{ f(x) + \frac{1}{2\rho}|x-x_k|^2 \right}. ]
Under sharp growth, one can show that if (x_{k+1}) is not optimal, then [ |x_{k+1}-x_k| \ge c ] for some constant (c>0).
Since the iterates remain in a bounded region, this cannot happen indefinitely.
Therefore the method terminates in finitely many steps.