A coupling of finite particle systems

We show that for a large class of one-dimensional interacting particle systems, with a finite initial configuration, any limit measure , for a sequence of times tending to infinity, must be invariant. This result is used to show that the one-dimensional biased annihilating branching process with parameter > 1/3 converges in distribution to the upper invariant measure provided its initial configuration is almost surely finite and non-null.