However the main point is that you can't generalize and give guarantees on a set of samples.
This is the world of Statistics. This is something completely different. AFAIK Statistics doesn't operate on sequences..
You can if it is natural or artificial evolution. Populations are Cauchy sequences. They converge to a single best individual (Cauchy) in the population, thus making it a complete metric space using the fitness function as a metric. aka Cauchy annealing schedule.
But some people don't believe in evolution.
Evolution is another example of a non-linear system that generates divergence, not convergence. It may be the best evidence of how dangerously wrong Cauchy sequences are for many categories of systems because on some timescales, the population of genotypes do seem to be converging toward "best" condition. But then something happens (a new competitor evolves, climate changes, species become toxic, disease sweeps the population, etc.) and whole chunks of the genome change quite quickly.
At the gene level, the "convergence model" is provably false. For example, the gene for sickle cell anemia causes debilitating problems if the person inherits two copies of it. And yet evolution will never drive this "bad mutation" out of the population because people who have no copies of this gene are more susceptible to dying of malaria. At best, the "equilibrium" condition is that all three genotypes persisting in the population. But even this equilibrium is highly unstable under the forces of evolution of both humans (who might evolve another defense against malaria or an adjunct that fixes the double-copy sickle cell problem) and the malaria parasite (which might evolve a counter measure to the sickle gene defense).
I could go on because there are quite a large number of phenomena that prevent convergence in evolution.