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

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The discussion centers on whether the superior limit of a sequence can be lower than its supremum. It is established that the supremum of a set of limits from subsequences may not equal the superior limit of the original sequence. An example provided illustrates this, where the sequence defined as xn = 1 + 1/n has a supremum of 2 and a superior limit of 1. This indicates that the superior limit can indeed be lower than the supremum. The conclusion emphasizes the distinction between these two concepts in sequence analysis.
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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!
 
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limsup may be lower than sup. Example xn = 1 + 1/n, sup = 2, limsup = 1.
 
Thanks, that was quite helpful.
 
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