Following the Convergence proof (on page 3) from Alternative Projection paper: https://web.stanford.edu/class/ee392o/alt_proj.pdf
I know intuitively how to show that both sequences {$ \left\lVert y_k - x_k \right\rVert_2$} and $ \left\lVert x_{k+1} - y_k \right\rVert_2$ converge to zero but I cannot show that formally in terms of equations. Any help will be appreciated.
If possible let $||x_k-y_k||_2$ not converge to 0. Hence $||x_k-y_k||_2 \ge \epsilon > 0$ for all $k$. Now we have from equations (1) and (2) that $$||x_{k+1}-\bar{x}||_2 \le ||y_k-x_k||_2 \le ||x_k-\bar{x}||_2 - \epsilon$$ But this cannot go on indefinitely because $||x_k-\bar{x}||_2$ is bounded below by $0$. Contradiction.