# Trying to understand lim sup

1. Feb 25, 2012

### 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!

2. Feb 27, 2012

### mathman

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

3. Feb 27, 2012

### IniquiTrance

Thanks, that was quite helpful.