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The totally asymmetric simple exclusion process (TASEP) is a basic model of statistical mechanics that has found numerous applications. We consider the case of TASEP with a finite chain where particles may enter from the left and leave to the right at prescribed rates. This model can be formulated as a Markov process with a finite number of states. Due to the irreducibility of the process it is well-known that the probability distribution on the states is globally attracted to a unique equilibrium distribution. We extend this result to the more detailed level of individual trajectories. To do so we formulate TASEP as a random dynamical system. Our main result is that the trajectories from all possible initial conditions contract to each other yielding the existence of a random attractor that consists of a single trajectory almost surely. This implies that in the long run TASEP ``filters out'' any perturbation that changes the state of the particles along the chain. In order to prove our main result we first establish that any random dynamical system on a finite state space possesses both a global random pullback attractor and a global random forward attractor. This observation appears to be missing in the literature. We then provide sufficient and necessary conditions for these attractors to be singletons. Finally, we show that TASEP satisfies one of these conditions.