# Proving a subsequence converges to S

• Sep 24th 2008, 01:32 PM
hockey777
Proving a subsequence converges to S
Let $(x_n)$ be a bounded sequence for each $n \in N$ let $s_n:=$sup{ $x_k:k$> $n$} and $S=inf${ $s_n$}. Show that there exists a sub sequence of $(x_n)$ that converges to S.

Ok so clearly $s_1$> $s_2$>....> $s_n$.

Since it is decreasing and bounded then $s_n$ converges to $S$. Futhermore there exists a subsequence of $s_n$ that is convergent that much also converge to S. My only problem is I don't know how to relate this back to $(x_n)$
• Sep 24th 2008, 01:51 PM
hockey777
I take that back, there is still need for help.
• Sep 24th 2008, 01:56 PM
ThePerfectHacker
Quote:

Originally Posted by hockey777
Ok so clearly $s_1$> $s_2$>....> $s_n$.

This does not make sense, $s_n$ are sets, how can they be "larger" or "smaller"?
• Sep 24th 2008, 02:00 PM
hockey777
Quote:

Originally Posted by ThePerfectHacker
This does not make sense, $s_n$ are sets, how can they be "larger" or "smaller"?

no $s_1$ is the first position in the sequence $s_n$, but I see where you think that as I messed up on my original post, I fixed it.