The discussion centers on the relationship between the supremum (sup) and the limit superior (limsup) of sequences, specifically in the context of countably infinite sets. Participants explore definitions, conditions under which these concepts are equal, and provide examples to illustrate their points.
Participants express some agreement on the definitions and relationships between sup and limsup, but there are also nuances and conditions that remain contested, particularly regarding the circumstances under which they are equal.
The discussion includes assumptions about boundedness and the nature of subsequences, which are not fully resolved. The implications of these assumptions on the definitions and relationships discussed are acknowledged but not clarified.
disregardthat said:limsup will equal sup if and only there is a subsequence converging to sup. In general sup is always larger or equal to limsup. Both are only well-defined if the sequence is bounded above.
limsup is the largest value for which there is a subsequence converging to it. In other words it's the largest limit point of the sequence. I may be mistaken, feel free to correct me if I'm wrong.
AxiomOfChoice said:No, this definitely makes sense. I suppose in the case of the sequence [itex]1, 1/2, 1/3, \ldots[/itex], we have [itex]\sup\limits_n x_n = 1[/itex] (since 1 is certainly the least upper bound), but [itex]\limsup\limits_{n\to \infty} x_n = 0[/itex] (since 0 is the only limit point of this set). Thanks!