This chapter is devoted to the proofs of the previously given theorems on pointwise convergence of Fourier series. The theorems of Dirichlet and Fejér with their implications are proven. The mitigation of the Gibbs phenomenon by summation kernels is shown as well as the Parseval equation for piecewise continuous periodic functions. As a summary of the acquired knowledge at that point, mathematical results are discussed, which have historically led from classical Fourier analysis and integration theory to the Lebesgue integral and distributions. The theoretical foundations of distribution theory and its countless practical applications are developed in the following chapters.