In　this　paper,　we　prove　some　analogues　of　Payne–Polya–Weinberger,　Hile–Protter　and　Yang’s　inequalities　for　Dirichlet　(discrete)　Laplace　eigenvalues　on　any　subset　in　the　integer　lattice　Zn　.　This　partially　answers　a　question　posed　by　Chung　and　Oden　(Pac　J　Math　192(2):257–273,　2000).　©　2023,　Mathematica　Josephina,　Inc.