This paper presents an elementary proof of the approximation of Sobolev spaces \( H^k(\Omega) \) by finite element functions using local regularization. The construction allows fitting boundary conditions in certain cases. The authors restrict their attention to triangular finite element subspaces and define a linear mapping \( \Pi \) that approximates functions in \( H^q(\Lambda) \) for \( q \leq \rho + 1 \). The error between the approximated function and the true function is shown to be bounded by \( c h^{q-k} |u|_{q, \Lambda} \), where \( h \) is the diameter of the elements in the decomposition. The paper also introduces a modified version \( \tilde{\Pi} \) that fits boundary conditions, specifically for functions that take the value 0 on the boundary \( \Gamma \). The proofs rely on several lemmas and hypotheses, including the existence of basis functions and the properties of the finite element spaces. The results are slightly less restrictive than those previously established by Strang and others.This paper presents an elementary proof of the approximation of Sobolev spaces \( H^k(\Omega) \) by finite element functions using local regularization. The construction allows fitting boundary conditions in certain cases. The authors restrict their attention to triangular finite element subspaces and define a linear mapping \( \Pi \) that approximates functions in \( H^q(\Lambda) \) for \( q \leq \rho + 1 \). The error between the approximated function and the true function is shown to be bounded by \( c h^{q-k} |u|_{q, \Lambda} \), where \( h \) is the diameter of the elements in the decomposition. The paper also introduces a modified version \( \tilde{\Pi} \) that fits boundary conditions, specifically for functions that take the value 0 on the boundary \( \Gamma \). The proofs rely on several lemmas and hypotheses, including the existence of basis functions and the properties of the finite element spaces. The results are slightly less restrictive than those previously established by Strang and others.