A generalised Lyapunov inequality is described which can be reformulated as nonstrict LMIs, for the check of regularity, impulse immunity, and stability of discrete-time descriptor systems simultaneously. Based on this inequality, a bounded real lemma in nonstrict LMIs is obtained which characterises properties of such descriptor systems, including regularity, impulse free property, stability, and H-infinity norm bound condition. The proofs are purely algebraic, therefore they are simple and definite. These results could play a key role in the LMI-based H-infinity controller design for discrete-time descriptor systems.