Subsequential Limit of a Sequence

[bphys]

I have a problem that asks for the subsequential limits, the limit superior, and the limit inferior for the sequence

$s_n = 4 ^{\frac {1} {n}}$

I haven't had trouble with my other problems, but I don't see any subsequences in the sequence (other than the sequence itself). Am I missing somthing?

 

[jbunniii]

I have a problem that asks for the subsequential limits, the limit superior, and the limit inferior for the sequence

$s_n = 4 ^{\frac {1} {n}}$

I haven't had trouble with my other problems, but I don't see any subsequences in the sequence (other than the sequence itself). Am I missing somthing?

All sequences have subsequences. For example, isn't $(s_{2n})$ a subsequence of $(s_n)$?

 

[HallsofIvy]

The "lim inf" of a sequence is the "infimum" (lower bound) of the set of all subsequential limits and the "lim sup" is the "supremum" (upper bound) of that same set. When you say you "can't find any subsequences" what you really mean is that you cannot find any subsequences that converge to different limits. That's not a problem. Because this sequence itself converges, it follows that all subsequences coverge to the same limit. "Limit inferior" and "limit superior" ("lim inf" and "lim sup") are both equal to that limit.