Example:
Is the set
{1, 1/2, 1/3, 1/4, ...} 
{0}
compact ?
This set is compact. To see this, first note that the set is bounded.
Hence, any sequence of elements from this set is also bounded, and
using the Bolzano-Weierstrass theorem we can extract a convergent
subsequence. But the set is also closed, so that by the theorem on
accumulation and boundary points the limit in fact must be part of
the set as well.
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