2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/170599The aim of this paper is to provide a proof for a version of Morse inequality for manifolds with boundary. Our main results are certainly known to the experts on Morse theory, nevertheless it seems necessary to write down a complete proof for it. Our proof is analytic and is based on J. Roe's account of Witten's approach to Morse Theory.A comment is added at the end of the Introduction.Differential GeometryAnalysis of PDEs58E05; 35J25text