This paper presents an elementary proof of a theorem on the approximation of Sobolev spaces $ H^q(\Omega) $ by finite elements without using classical interpolation. The construction allows fitting boundary conditions in some cases. The paper focuses on triangular finite element subspaces and provides estimates for the error in terms of Sobolev norms. The main result is that for $ u \in H^q(\Lambda) $, $ q \leq \rho + 1 $, the approximation error $ |u - \Pi u|_{k,\Lambda} \leq c h^{q-k} |u|_{q,\Lambda} $ for $ k = 0, 1, ..., q $, and if $ q \leq \rho $, the error tends to zero as $ h \to 0 $. The paper also discusses a modified version of the approximation to fit boundary conditions, showing similar error estimates. The results are based on hypotheses regarding the properties of the finite element space and the domain. The proofs rely on lemmas that establish bounds on the approximation error and use properties of Sobolev spaces and finite elements. The paper concludes with remarks on the applicability of the results to one-dimensional finite elements and other variations of the approximation method.This paper presents an elementary proof of a theorem on the approximation of Sobolev spaces $ H^q(\Omega) $ by finite elements without using classical interpolation. The construction allows fitting boundary conditions in some cases. The paper focuses on triangular finite element subspaces and provides estimates for the error in terms of Sobolev norms. The main result is that for $ u \in H^q(\Lambda) $, $ q \leq \rho + 1 $, the approximation error $ |u - \Pi u|_{k,\Lambda} \leq c h^{q-k} |u|_{q,\Lambda} $ for $ k = 0, 1, ..., q $, and if $ q \leq \rho $, the error tends to zero as $ h \to 0 $. The paper also discusses a modified version of the approximation to fit boundary conditions, showing similar error estimates. The results are based on hypotheses regarding the properties of the finite element space and the domain. The proofs rely on lemmas that establish bounds on the approximation error and use properties of Sobolev spaces and finite elements. The paper concludes with remarks on the applicability of the results to one-dimensional finite elements and other variations of the approximation method.