2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/163040We give an elementary proof of Grothendieck's non-vanishing Theorem: For a finitely generated non-zero module $M$ over a Noetherian local ring $A$ with maximal ideal $\m$, the local cohomology module $H^{\dim M}_{\m}(M)$ is non-zero.Title & abstract changed, some minor changes in the main body of the paper, 3 pages, To appear in Communications in AlgebraCommutative AlgebraAlgebraic Geometry13D45, 14B15 (Primary), 13A30 (Secondary)text