This article investigates a high-order convergent multistep method for solving nonlinear equations, with particular emphasis on the method's local and semi-local convergence properties in general Banach space settings. Conventional convergence analyses typically rely on Taylor series expansions and require the computation of higher-order derivatives, which are not part of the method. Such schemes may be costly, impractical, or nonexistent and are restricted to Euclidean spaces of finite dimension. Other constraints of such analyses involve the absence of a priori error estimates as well as uniqueness of the solution results. This is why, in this article, the convergence is established using only the operators of the method in combination with the concept of generalized continuity required to control the derivative. This is how the applicability of the method is extended. Due to its generality the same methodology is applicable to other methods. Numerical examples are provided to support and illustrate the theoretical findings.