A speed-limited robot travels in a dynamic environment cluttered with arbitrarily shaped moving obstacles. There also is an unpredictably moving target in the scene. The robot measures the heading angle to the target and the distance to the nearest obstacle along any ray emitted from the robot. A simple reactive navigation strategy is presented that autonomously drives the robot to the target in a finite time provided that the algorithm is properly tuned. This is shown via a mathematically rigorous global convergence result under minor and partly unavoidable technical assumptions. Feasible closed-form recommendations on controller tuning are offered. Theoretical results are confirmed by computer simulation tests.