Prove the convergence of successive convex approximation

I recently saw an paper that performed successive convex approximation operations on a non-convex problem, but the proof of its convergence is very simple. (The specific situation is shown in the figure)

Please help me to see if this algorithm can give a strict mathematical proof of convergence?

Thank you very much1


Any proof of convergence of Successive Convex Approximation is out of scope of this forum. Perhaps a much better formulated and more specific question at or might get an answer.

Thanks for your suggestion!