Incremental Neural Networks (INNs) serve as fundamental instruments for attaining highly accurate approximation outcomes. Specifically, convexity constitutes a crucial property in the context of approximating neural networks within incremental frameworks. This characteristic facilitates the derivation of precise approximations of the target function. Moreover, convexity is applicable to both constrained (i.e., convex) and unconstrained approximation scenarios. Furthermore, leveraging convexity in both network architectures and approximation processes enables the determination of optimal function approximations through incremental neural networks. In the present study, we perform an approximation of neural networks by driving the difference sequence between the target function and the Modulus of Convexity in Incremental Neural Networks (MCINNs) toward zero. Consequently, we also derive the degree of approximation via neural network approximation, yielding a rate of O(1/n^(p+1) ). These theoretical findings are corroborated by experimental results obtained using MATLAB, wherein an algorithm initialized with the neural network was implemented. Additionally, the algorithm demonstrates, for several test functions, an improved approximation quality compared to the standalone neural network. The convex incremental iteration algorithm progressively approximates the objective function by means of a neural network, relying on a sequence of functions to achieve enhanced approximation performance. The residual error was compared against the Mean Squared Error (MSE), resulting in a lower residual error value.