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In this paper, we describe a novel approach for checking safety specifications of a dynamical system with exogenous inputs over infinite time horizon. We introduce the notion of resolution completeness for analysis of safety falsification algorithms and present sampling-based resolution-complete algorithms for safety falsification of discrete-time linear time-invariant systems. Given a target resolution of inputs, the algorithms terminate either with a reachable state that violates the safety specification, or prove that the system does not violate the specification at the given resolution of inputs.