Definition 3.3.1 (Compactness). A set K ⊆ R is compact if every sequence in K has a subsequence that converges to a limit that is also in K.
Decide which of the following sets are compact. For those that are not compact, show how Definition 3.3.1 breaks down. In other words, give an example of a sequence contained in the given set that does not possess a subsequence converging to a limit in the set.
(a) Z