In Euclidean space, closed and bounded is exactly sequential compactness.Suggest a slogan
Let \(n\geq1\) and \(K\subseteq\RR^n\). Then \(K\) is closed and bounded if and only if every sequence in \(K\) has a subsequence converging to a point of \(K\). In particular, every bounded sequence in \(\RR^n\) has a convergent subsequence.