Consider the following minimization problem: $$\min_{x\in \mathbb{R}^n}{f(x)},$$ where f is a proper convex closed function such that there exists at least one solution of the above-mentioned problem. Proximal point algorithm works choosing an initial point $x_0$ and the minimizing sequence is defined recursively by $$x_k=\arg\,\min_{x\in \mathbb{R}^n}{f(x)+\frac{1}{2a_k}{\lVert x-x_{k-1}\rVert}}^2,\quad \text{for any }k\in \mathbb{N}.$$
In Tikhonov regularization the $n$-th term of the minimizing sequence is defined by $$\bar{x}_k=\arg\,\min_{x\in \mathbb{R}^n}{f(x)+\frac{1}{2a_k}\lVert x\rVert}^2, \quad \text{for any }k\in \mathbb{N}.$$
My question is: if I choose $x_0=0$ in proximal algorithm, does the limit of the sequence $(x_k)_k$ coincide with the limit of $(\bar{x}_k)_k$?
I suppose not but I cannot provide an example.