Understanding FISTA's weak convergence: A step-by-step introduction to the 2025 milestone

Beck and Teboulle’s FISTA for finding the minimizer of the sum of two convex functions is one of the most important algorithms of the past decades. While function value convergence of the iterates was known, the actual convergence of the iterates remained elusive until October 2025 when Jang and Ryu, as well as Boţ, Fadili, and Nguyen proved weak convergence. In this paper, we provide a gentle self-contained introduction to the proof of their remarkable result.