Prove the convergence of successive convex approximation

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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

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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 https://or.stackexchange.com/ or https://math.stackexchange.com/ might get an answer.

3

Thanks for your suggestion!