Physicists don't use Cauchy sequences it seems.
"As to the axioms of the theory of probabilities, it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases. ... Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua."
Interesting!
Perhaps this is because the population total of a physical quantity (mass, energy, velocity, charge, etc.) is bounded and therefore the per-particle or per-unit average is bounded, too. (But if the universe is infinite then.....)
Does Cauchy ever occur in the real world or is it all the minds of mathematicians?