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This paper is concerned with the stability of rational one-step approximations of C 0 semigroups. Particular emphasis is laid on long-term stability bounds. The analysis is based on a general Banach space framework and allows variable stepsize sequences. Under reasonable assumptions on the stepsize sequence, asymptotic stability bounds for general C 0 semigroups are derived. The bounds are typical in the sense that they contain, in general, a factor that grows with the number of steps. Underdoi:10.1090/s0025-5718-01-01389-8 fatcat:ke6wfwsztvaeresdvet6ggfpvm