Can someone give an elegant proof to the following question?
For any sequence, there exist a monotonic subsequence whose limit is $\limsup S_n$ and there exists a monotonic subsequence whose limit is $\liminf S_n$.
Can someone give an elegant proof to the following question?
For any sequence, there exist a monotonic subsequence whose limit is $\limsup S_n$ and there exists a monotonic subsequence whose limit is $\liminf S_n$.
Since $S:=\limsup x_n$ is a cluster point of the real sequence $x=(x_n)$, there exists a subsequence of $x$ which converges to $S$.
Using the fact that every real sequence has a monotone subsequence you obtain the wanted result.
You can find proofs of both results here (and probably in many other texts).