Let $\displaystyle (x_n)$ be a bounded sequence for each $\displaystyle n \in N$ let $\displaystyle s_n:=$sup{$\displaystyle x_k:k$>$\displaystyle n$} and $\displaystyle S=inf${$\displaystyle s_n$}. Show that there exists a sub sequence of $\displaystyle (x_n)$ that converges to S.

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

Since it is decreasing and bounded then $\displaystyle s_n$ converges to $\displaystyle S$. Futhermore there exists a subsequence of $\displaystyle s_n$ that is convergent that much also converge to S. My only problem is I don't know how to relate this back to $\displaystyle (x_n)$