Abstract: Based on the Element Energy Projection (EEP) method, an improved scheme with optimal order of super-convergence, is presented for one-dimensional C⁰ FEM, i.e., FEM sulotions can be obtained through the scheme for the elements with sufficient smooth property and m degrees. The proposed scheme is capable of producingO(h^2m) super-convergence for both displacements and stresses at any point on an element in post-processing stage. The entire work is composed of three parts, i.e. formulation, numerical results as well as mathematical analysis. The present paper is the third in the series and gives the mathematical proof of the optimalO(h^2m) super-convergence for the proposed scheme.