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We address the discrete-time lion and man problem in a bounded, convex, planar environment in which both players have identical sensing ranges, restricted to closed discs about their current locations. The evader is randomly located inside the environment and moves only when detected. The players can step inside identical closed discs, centered at their respective positions. We propose a sweep-pursuit-capture strategy for the pursuer to capture the evader. The sweep phase is a search operation by the pursuer to detect an evader within its sensing radius. In the pursuit phase, the pursuer employs a greedy strategy of moving towards the last-sensed evader position. We show that in finite time, the problem reduces to a previously-studied problem with unlimited sensing, which allows us to use the established lion strategy in the capture phase. We give a novel upper bound on the time required for the pursuit phase to terminate using the greedy strategy and a sufficient condition for this strategy to work in terms of the value of the ratio of sensing to stepping radius of the players.