What is the relationship between lim sup and the limit of a subsequence?

[filter54321]

Homework Statement

Show that the lim sup of a bounded sequence is a limit of a subsequence.

Homework Equations

Sequence: S_n
Subsequence: S_nk

The Attempt at a Solution

An existent lim sup means that at a large enough N, the subsequence could hug the bottom of the lim sup to within epsilon (e). I don't know how to formalize this notion.

The last two steps of the proof are likely
-e < S_nk - lim sup S_n < e
|S_nk - lim sup S_n| < e

 

[HallsofIvy]

The lim sup is the sup of the set of all subsequential limits. That means that, calling the lim sup a, given any $\epsilon> 0$ there exist a subsequential limit within $\epsilon$ of a (and less than a). In particular, for every positive integer m, there exist a subsequential limit within 1/2m of a. And because that is a limit of a subsequence, there exist a member of that subsequence, call it $a_{m}$, within 1/2m of that subsequential limit. Look at what the subsequence $a_m$ converges to. You are forming a new subsequence by, essentially, taking a member from every subsequence.