Heine–Borel theorem
The Heine–Borel theorem is the bridge between the abstract cover-based definition of compactness and the concrete picture we carry from calculus. It says that in ordinary Euclidean space the elusive condition “every open cover has a finite subcover” simplifies to two things you can check at a glance: the set is closed, and it is bounded.
Precisely: a subset of R^n is compact if and only if it is closed and bounded. So a closed disk, a closed ball, the surface of a sphere, or the interval [a, b] are all compact, whereas an open interval (it is not closed), the whole line (it is not bounded), or the set of integers (closed but unbounded) are not.
The theorem is special to finite-dimensional Euclidean space — its real content is the hard direction, that closed-and-bounded forces the finite-subcover property. In a general metric space closed and bounded is necessary but not sufficient: the closed unit ball of an infinite-dimensional Banach space is closed and bounded yet famously not compact. So do not export “compact = closed and bounded” beyond R^n; the correct general criterion is closed and totally bounded (in a complete space).
The forward direction (compact implies closed and bounded) holds in any metric space and is easy. It is the reverse direction that needs R^n's finite dimensionality, ultimately resting on completeness plus the fact that bounded closed boxes are compact (a consequence of the nested-interval / Bolzano–Weierstrass machinery).