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!
The Sup (supremum) of a sequence is the smallest number that is greater than or equal to all the terms in the sequence. The Limsup (limit superior) is the largest limit point of a sequence, meaning it is the largest number that a subsequence of the sequence can approach.
The Limsup is always greater than or equal to the Sup of a sequence. This is because the Limsup is the largest limit point and the Sup is the smallest number that is greater than or equal to all the terms in the sequence.
The Sup and Limsup of Sequences are important concepts in real analysis and measure theory. They are used to define the convergence and divergence of sequences, as well as to prove theorems in these areas of mathematics.
If the Sup and Limsup of a sequence are equal, then the sequence is convergent. If the Sup is greater than the Limsup, then the sequence is divergent. If the Sup is equal to infinity, then the sequence is unbounded.
Yes, Sup and Limsup of Sequences have practical applications in fields such as statistics, engineering, and physics. They are used to analyze data trends, model systems, and make predictions about outcomes.