Name one such sequence of "useful reals" that is Cauchy but doesn't converge to a "useful real". You can't, can you? "Useful" Cauchy sequences of "useful reals"

Name one such sequence of "useful reals" that is Cauchy but doesn't converge to a "useful real". You can't, can you? "Useful" Cauchy sequences of "useful reals" (i.e. those you can define) all converge to a "useful real".