On 21 Nov., 16:24, William Hughes <wpihug...@gmail.com> wrote: > Your basic problem remains. You continue to talk > about "the" limit as if there was only one.
The real sequence has a limit. And if you dislike infinity as an improper limit, then take the reciprocals. They have *the* limit 0. This sequence is independent of anything else but its terms or its definition.
Set theory shows that *this sequence* has a limit without indices on the left hand side, and hence has another limit (< 1) or no limit. Or the reciprocals have a limit > 0. This result does in no way depend on anything else but set theory being incompatible with mathematics.