Can the Superior Limit of a Sequence Be Lower Than Its Supremum?

[IniquiTrance]

Suppose there is some sequence $\{x_n\}_{n\in\mathbb{N}}$. Say we have a set of all the limits of all possible subsequences, would the supremum of this set be the superior limit of $\{x_n\}_{n\in\mathbb{N}}$? What about if this value turns out to be 5, but there is a member of the sequence that is equal to 500, but is not the limit of any subsequence. Can the superior limit be lower than the supremum of the sequence? Thanks!

 

[mathman]

limsup may be lower than sup. Example x_n = 1 + 1/n, sup = 2, limsup = 1.

 

[IniquiTrance]

Thanks, that was quite helpful.