If $E$ iis a compact, nonempty subset of REAL numbers (hence closed), we know that every convergent sequence in $E$ converges in $E$, are there sequences $a_{n}$ and $b_{n}$ in $E$ that converge to $\sup E$ and $\inf E$, respectively?
I would think so since we can just take any collection on points in $E$ that are arbitrary close to $E$, but how would I show this, or is it not true in an arbitrary metric space?