Definition 3.3.1 (Compactness). A set \(K \subseteq \mathbf{R}\) is compact if every sequence in \(K\) has a subsequence that converges to a limit that is also in \(K\).


Example

A closed interval \([c,d]\) is a compact set. To see why, notice that if a sequence \((a_n)\) is contained in \([c,d]\) then we can use the Bolzano-Weirstrass theorem which states that every bounded sequence contains a convergent subsequence to conclude that we are guaranteed to find a convergent subsequence \((a_{n_k})\). We also know from this, that every closed interval is a closed set. Therefore, \([c,d]\) contains all of its limit points and the limit of \((a_{n_k})\) is in \([c,d]\) as required.

Other Definitions and Properties


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