BoostConv has been introduced in earlier works as an effective acceleration technique for nonlinear iterative processes and has been successfully employed in a variety of applications to enhance convergence rates or to compute unstable fixed points that are otherwise inaccessible through standard approaches. Despite its demonstrated practical effectiveness, the theoretical properties of the method have not yet been fully characterized. In this work, we present a more robust formulation of the BoostConv algorithm and, for the first time, provide a rigorous proof of its convergence. The proposed analysis places BoostConv within a precise mathematical framework, clarifying its interpretation as a nonlinear convergence accelerator and establishing sufficient conditions under which convergence to a fixed point is guaranteed. The theoretical findings are illustrated through several numerical examples, spanning from a linear problem to a low-dimensional benchmark and a large-scale incompressible Navier-Stokes simulation. These results demonstrate the robustness and practical relevance of the proposed method and bridge the gap between empirical performance and rigorous analysis, paving the way for further developments and applications to complex nonlinear problems.