Sorry if I misunderstand you. If I have a Cauchy sequence of "useful reals", wouldn't the convergence be, by definition, a "useful real"? That is, I can write down the Cauchy sequence, so it's now symbolically noted, right?
Or are you referring to a Cauchy sequence that exists, but can't be defined using our symbology?
Or are you referring to a Cauchy sequence that exists, but can't be defined using our symbology?